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

Goal state before:
pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ ≤ π
a✝ : Nontrivial A
⊢ Real.cos ‖x‖ ∈ lowerBounds ((fun a ↦ Real.cos a) '' spectrum ℝ ↑x)

Tactic:
·

Goal state after:
case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ ≤ π
a✝ : Nontrivial A
⊢ Real.cos ‖x‖ ∈ lowerBounds ((fun a ↦ Real.cos a) '' spectrum ℝ ↑x)
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
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h✝² : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝³ : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝² : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝¹ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝¹ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this✝ : Nonempty E
e : ↑(f ⁻¹' {x})
s : Set X
hs : s ⊆ U'
h✝ : IsOpen[inst✝²] (f ⁻¹' s ∩ V' e)
this : s ⊆ (φ e).target
e' : E
h : e' ∈ f ⁻¹' s ∩ (φ e).source
e'' : E
left✝ : e'' ∈ V ↑e
mem : e'' ∈ (φ e).source
eq : f e'' = f e'
⊢ e' ∈ V ↑e

Tactic:
← (φ e).injOn mem h.2 (by rwa [(hφ e).2])

Goal state after:
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h✝² : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝³ : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝² : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝¹ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝¹ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this✝ : Nonempty E
e : ↑(f ⁻¹' {x})
s : Set X
hs : s ⊆ U'
h✝ : IsOpen[inst✝²] (f ⁻¹' s ∩ V' e)
this : s ⊆ (φ e).target
e' : E
h : e' ∈ f ⁻¹' s ∩ (φ e).source
e'' : E
left✝ : e'' ∈ V ↑e
mem : e'' ∈ (φ e).source
eq : f e'' = f e'
⊢ e'' ∈ V ↑e
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: case inr.refine_3
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝² : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝¹ : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this : Nonempty E
e : ↑(f ⁻¹' {x})
⊢ InjOn f (V' e)

Tactic:
·

Goal state after:
case inr.refine_3
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝² : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝¹ : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this : Nonempty E
e : ↑(f ⁻¹' {x})
⊢ InjOn f (V' e)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case refine_1
n : ℤ
inst✝ : NeZero n
hn : IsUnit ↑n
e : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯
⊢ Topology.IsQuotientMap ⇑(zpowGroupHom n)

pp: case refine_2
n : ℤ
inst✝ : NeZero n
hn : IsUnit ↑n
e : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯
⊢ (⇑e ⁻¹' ↑(zpowGroupHom n).ker).Finite

Tactic:
·

Goal state after:
case refine_1
n : ℤ
inst✝ : NeZero n
hn : IsUnit ↑n
e : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯
⊢ Topology.IsQuotientMap ⇑(zpowGroupHom n)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case refine_2
n : ℤ
inst✝ : NeZero n
hn : IsUnit ↑n
e : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯
⊢ (⇑e ⁻¹' ↑(zpowGroupHom n).ker).Finite

Tactic:
·

Goal state after:
case refine_2
n : ℤ
inst✝ : NeZero n
hn : IsUnit ↑n
e : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯
⊢ (⇑e ⁻¹' ↑(zpowGroupHom n).ker).Finite
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
inst✝ : T2Space E
hf : IsClosedMap f
hs : ∀ x ∈ s, (f ⁻¹' {x}).Finite
h : IsLocalHomeomorphOn f (f ⁻¹' s)
⊢ IsCoveringMapOn f s

Tactic:
by
  intro x hx
  refine hf.isEvenlyCovered_of_openPartialHomeomorph (hs x hx) fun e he ↦ ?_
  obtain ⟨φ, hφ, rfl⟩ := h e (by aesop)
  aesop

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Normalize

Goal state before:
pp: case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
hx : x = 0
⊢ normalize (normalize x) = normalize x

pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
hx : ¬x = 0
⊢ normalize (normalize x) = normalize x

Tactic:
·

Goal state after:
case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
hx : x = 0
⊢ normalize (normalize x) = normalize x
Module: Mathlib.Analysis.Normed.Module.Normalize

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
hx : ¬x = 0
⊢ normalize (normalize x) = normalize x

Tactic:
·

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
hx : ¬x = 0
⊢ normalize (normalize x) = normalize x
Module: Mathlib.Analysis.Normed.Module.Normalize

Goal state before:
pp: case inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
h_pos : 0 < r
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x

pp: case inr.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
⊢ normalize (0 • x) = ↑(SignType.sign 0) • normalize x

pp: case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
h_neg : r < 0
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x

Tactic:
·

Goal state after:
case inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
h_pos : 0 < r
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x
Module: Mathlib.Analysis.Normed.Module.Normalize

Goal state before:
pp: case inr.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
⊢ normalize (0 • x) = ↑(SignType.sign 0) • normalize x

pp: case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
h_neg : r < 0
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x

Tactic:
·

Goal state after:
case inr.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
x : V
⊢ normalize (0 • x) = ↑(SignType.sign 0) • normalize x
Module: Mathlib.Analysis.Normed.Module.Normalize

Goal state before:
pp: case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
h_neg : r < 0
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x

Tactic:
·

Goal state after:
case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace ℝ V
r : ℝ
x : V
h_neg : r < 0
⊢ normalize (r • x) = ↑(SignType.sign r) • normalize x
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsSelfAdjoint a
ha_norm : ‖a‖ ≤ 1
h✝ : Subsingleton A
⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A

pp: case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsSelfAdjoint a
ha_norm : ‖a‖ ≤ 1
h✝ : Nontrivial A
⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A

Tactic:
·

Goal state after:
case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsSelfAdjoint a
ha_norm : ‖a‖ ≤ 1
h✝ : Subsingleton A
⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
y : ℂ
hy : y ∈ spectrum ℂ ↑u
⊢ ‖↑y.arg‖ ≤ π

Tactic:
by simpa using abs_arg_le_pi y

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

Goal state before:
pp: case inl
A : Type u_1
inst✝ : CStarAlgebra A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
⊢ ∃ u c, 0 = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖0‖ / 2

pp: case inr
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
⊢ ∃ u c, x = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖x‖ / 2

Tactic:
·

Goal state after:
case inl
A : Type u_1
inst✝ : CStarAlgebra A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
⊢ ∃ u c, 0 = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖0‖ / 2
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 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‖

pp: 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‖

Tactic:
·

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‖
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
t : ℝ
ht : t ∈ Set.Icc 0 1
hu : ‖↑u - 1‖ < 2
key : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖
⊢ 2 * (1 - Real.cos ‖t • argSelfAdjoint u‖) ≤ 2 * (1 - Real.cos ‖argSelfAdjoint u‖)

pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
t : ℝ
ht : t ∈ Set.Icc 0 1
hu : ‖↑u - 1‖ < 2
key : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖
⊢ 2 * (1 - Real.cos ‖argSelfAdjoint u‖) ≤ ‖↑u - 1‖ ^ 2

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
t : ℝ
ht : t ∈ Set.Icc 0 1
hu : ‖↑u - 1‖ < 2
key : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖
⊢ 2 * (1 - Real.cos ‖t • argSelfAdjoint u‖) ≤ 2 * (1 - Real.cos ‖argSelfAdjoint u‖)
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: case inr
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
⊢ ∃ u c, x = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖x‖ / 2

Tactic:
·

Goal state after:
case inr
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type ?u.41) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
⊢ ∃ u c, x = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖x‖ / 2
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 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‖

Tactic:
·

Goal state after:
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.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
t : ℝ
ht : t ∈ Set.Icc 0 1
hu : ‖↑u - 1‖ < 2
key : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖
⊢ 2 * (1 - Real.cos ‖argSelfAdjoint u‖) ≤ ‖↑u - 1‖ ^ 2

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
t : ℝ
ht : t ∈ Set.Icc 0 1
hu : ‖↑u - 1‖ < 2
key : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖
⊢ 2 * (1 - Real.cos ‖argSelfAdjoint u‖) ≤ ‖↑u - 1‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: case h.left
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type u_1) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type u_1) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
u₁ : ↥(unitary A) := ⋯
u₂ : ↥(unitary A) := ⋯
this : x = ‖x‖ • 2⁻¹ • (↑u₁ + ↑(star u₁) + I • (↑u₂ + ↑(star u₂)))
⊢ x = ∑ i, ![↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹ * I, ↑‖x‖ * 2⁻¹ * I] i • ↑(![u₁, star u₁, u₂, star u₂] i)

pp: case h.right
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type u_1) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type u_1) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
u₁ : ↥(unitary A) := ⋯
u₂ : ↥(unitary A) := ⋯
this : x = ‖x‖ • 2⁻¹ • (↑u₁ + ↑(star u₁) + I • (↑u₂ + ↑(star u₂)))
⊢ ∀ (i : Fin 4), ‖![↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹ * I, ↑‖x‖ * 2⁻¹ * I] i‖ ≤ ‖x‖ / 2

Tactic:
·

Goal state after:
case h.left
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type u_1) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type u_1) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
u₁ : ↥(unitary A) := ⋯
u₂ : ↥(unitary A) := ⋯
this : x = ‖x‖ • 2⁻¹ • (↑u₁ + ↑(star u₁) + I • (↑u₂ + ↑(star u₂)))
⊢ x = ∑ i, ![↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹ * I, ↑‖x‖ * 2⁻¹ * I] i • ↑(![u₁, star u₁, u₂, star u₂] i)
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ Subtype.val '' closedBall 1 √ε ⊆ {a | IsStarNormal a ∧ spectrum ℂ a ⊆ sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε}}

pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ IsCompact (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})

pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (fun x ↦ ↑x.arg) (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ Subtype.val '' closedBall 1 √ε ⊆ {a | IsStarNormal a ∧ spectrum ℂ a ⊆ sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε}}
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
v : ↥(unitary A)
hv : v ∈ closedBall 1 √ε
z : ℂ
hz : z ∈ spectrum ℂ ↑v
⊢ 2 * (1 - z.re) ≤ ‖↑v - 1‖ ^ 2

pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
v : ↥(unitary A)
hv : v ∈ closedBall 1 √ε
z : ℂ
hz : z ∈ spectrum ℂ ↑v
⊢ ‖↑v - 1‖ ^ 2 ≤ ε

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
v : ↥(unitary A)
hv : v ∈ closedBall 1 √ε
z : ℂ
hz : z ∈ spectrum ℂ ↑v
⊢ 2 * (1 - z.re) ≤ ‖↑v - 1‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
v : ↥(unitary A)
hv : v ∈ closedBall 1 √ε
z : ℂ
hz : z ∈ spectrum ℂ ↑v
⊢ ‖↑v - 1‖ ^ 2 ≤ ε

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
v : ↥(unitary A)
hv : v ∈ closedBall 1 √ε
z : ℂ
hz : z ∈ spectrum ℂ ↑v
⊢ ‖↑v - 1‖ ^ 2 ≤ ε
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: case h.right
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type u_1) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type u_1) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
u₁ : ↥(unitary A) := ⋯
u₂ : ↥(unitary A) := ⋯
this : x = ‖x‖ • 2⁻¹ • (↑u₁ + ↑(star u₁) + I • (↑u₂ + ↑(star u₂)))
⊢ ∀ (i : Fin 4), ‖![↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹ * I, ↑‖x‖ * 2⁻¹ * I] i‖ ≤ ‖x‖ / 2

Tactic:
·

Goal state after:
case h.right
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type u_1) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type u_1) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
u₁ : ↥(unitary A) := ⋯
u₂ : ↥(unitary A) := ⋯
this : x = ‖x‖ • 2⁻¹ • (↑u₁ + ↑(star u₁) + I • (↑u₂ + ↑(star u₂)))
⊢ ∀ (i : Fin 4), ‖![↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹ * I, ↑‖x‖ * 2⁻¹ * I] i‖ ≤ ‖x‖ / 2
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ IsCompact (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})

pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (fun x ↦ ↑x.arg) (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ IsCompact (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (fun x ↦ ↑x.arg) (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})

Tactic:
·

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ ContinuousOn (fun x ↦ ↑x.arg) (sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε})
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
B : F →L[𝕜] E
⊢ A = adjoint B ↔ ∀ (x : E) (y : F), ⟪A x, y⟫_𝕜 = ⟪x, B y⟫_𝕜

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

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
A : E →L[𝕜] F
B : F →L[𝕜] E
h : ∀ (x : E) (y : F), ⟪A x, y⟫_𝕜 = ⟪x, B y⟫_𝕜
⊢ A = adjoint B
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: case mp
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
⊢ u ∈ pathComponent 1 → ∃ l, (List.map expUnitary l).prod = u

pp: case mpr
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
⊢ (∃ l, (List.map expUnitary l).prod = u) → u ∈ pathComponent 1

Tactic:
·

Goal state after:
case mp
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
⊢ u ∈ pathComponent 1 → ∃ l, (List.map expUnitary l).prod = u
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: case mpr
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
⊢ (∃ l, (List.map expUnitary l).prod = u) → u ∈ pathComponent 1

Tactic:
·

Goal state after:
case mpr
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
⊢ (∃ l, (List.map expUnitary l).prod = u) → u ∈ pathComponent 1
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: case a
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
x : E
hx : x ∈ (↑(adjoint T)).rangeᗮ
⊢ x ∈ (↑T).ker

pp: case a
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
x : E
hx : x ∈ (↑T).ker
⊢ x ∈ (↑(adjoint T)).rangeᗮ

Tactic:
·

Goal state after:
case a
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
x : E
hx : x ∈ (↑(adjoint T)).rangeᗮ
⊢ x ∈ (↑T).ker
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: case a
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
x : E
hx : x ∈ (↑T).ker
⊢ x ∈ (↑(adjoint T)).rangeᗮ

Tactic:
·

Goal state after:
case a
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
x : E
hx : x ∈ (↑T).ker
⊢ x ∈ (↑(adjoint T)).rangeᗮ