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

Goal state before:
pp: case h.refine_2
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
g : C(X, ℝ)
hgs : EqOn (⇑g) 0 s
hgt : EqOn (⇑g) 1 t
hicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1
⊢ EqOn (⇑(1 - g)) 1 s

Tactic:
intro x hx

Goal state after:
case h.refine_2
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
g : C(X, ℝ)
hgs : EqOn (⇑g) 0 s
hgt : EqOn (⇑g) 1 t
hicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1
x : X
hx : x ∈ s
⊢ (1 - g) x = 1 x
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_4
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 ∈ t, ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x

Tactic:
intro x hx

Goal state after:
case refine_4
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
hx : x ∈ t
⊢ ∃ t_1 ∈ nhdsWithin x t, ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t_1, 0 < g' x
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
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
⊢ EqOn g 0 mᶜ

Tactic:
intro x hx

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
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
x : X
hx : x ∈ mᶜ
⊢ g x = 0 x
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
⊢ ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t

Tactic:
have : ∃ g' : C(X, ℝ≥0), (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x := by
    refine ht.isCompact.induction_on ?_ ?_ ?_ ?_
    · refine ⟨0, ?_, fun x hx => False.elim hx⟩
      convert! I.zero_mem
      ext
      simp only [comp_apply, zero_apply, ContinuousMap.coe_coe, map_zero]
    · rintro s₁ s₂ hs ⟨g, hI, hgt⟩; exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩
    · rintro s₁ s₂ ⟨g₁, hI₁, hgt₁⟩ ⟨g₂, hI₂, hgt₂⟩
      refine ⟨g₁ + g₂, ?_, fun x hx => ?_⟩
      · convert! I.add_mem hI₁ hI₂
        ext y
        simp
      · rcases hx with (hx | hx)
        · simpa using add_lt_add_of_lt_of_le (hgt₁ x hx) zero_le
        · simpa using add_lt_add_of_le_of_lt zero_le (hgt₂ x hx)
    · intro x hx
      replace hx := htI.subset_compl_right hx
      rw [compl_compl, mem_setOfIdeal] at hx
      obtain ⟨g, hI, hgx⟩ := hx
      have := (map_continuous g).continuousAt.eventually_ne hgx
      refine
        ⟨{y : X | g y ≠ 0} ∩ t,
          mem_nhdsWithin_iff_exists_mem_nhds_inter.mpr ⟨_, this, Set.Subset.rfl⟩,
          ⟨⟨fun x => ‖g x‖₊ ^ 2, (map_continuous g).nnnorm.fun_pow 2⟩, ?_, fun x hx =>
            pow_pos (norm_pos_iff.mpr hx.1) 2⟩⟩
      convert! I.mul_mem_left (star g) hI
      ext
      simp only [comp_apply, coe_mk, algebraMapCLM_apply, map_pow,
        mul_apply, star_apply, star_def]
      simp only [RCLike.conj_mul]
      rfl

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
this : ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x
⊢ ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t
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
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := ⋯
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
⊢ sᶜ ⊆ (⇑{ toFun := g, continuous_toFun := ⋯ } ⁻¹' {1})ᶜ

Tactic:
intro x hx

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
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
x : X
hx : x ∈ sᶜ
⊢ x ∈ (⇑{ toFun := g, continuous_toFun := ⋯ } ⁻¹' {1})ᶜ
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : CommCStarAlgebra A
a : A
this : spectralRadius ℂ ((gelfandTransform ℂ A) (star a) * (gelfandTransform ℂ A) a) = ↑‖star a * a‖₊
⊢ ‖(gelfandTransform ℂ A) a‖ = ‖a‖

Tactic:
gelfandTransform_map_star,

Goal state after:
A : Type u_1
inst✝ : CommCStarAlgebra A
a : A
this : spectralRadius ℂ (star ((gelfandTransform ℂ A) a) * (gelfandTransform ℂ A) a) = ↑‖star a * a‖₊
⊢ ‖(gelfandTransform ℂ A) a‖ = ‖a‖
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ EqOn c.lim 1 t

Tactic:
intro x hx

Goal state after:
case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
x : X
hx : x ∈ t
⊢ c.lim x = 1 x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
r : ℝ
hr : 0 ≤ r
a✝ : Nontrivial A
⊢ ∀ x ∈ spectrum ℝ (r • a), 0 ≤ x

Tactic:
intro x hx

Goal state after:
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
r : ℝ
hr : 0 ≤ r
a✝ : Nontrivial A
x : ℝ
hx : x ∈ spectrum ℝ (r • a)
⊢ 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -0 = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
neg_zero,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : 0 = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
a : A
y : ℝ≥0
ha : 0 ≤ a
⊢ Set.EqOn (fun x ↦ ↑(x.toNNReal.nnrpow y)) (fun x ↦ x ^ ↑y) (quasispectrum ℝ a)

Tactic:
intro x hx

Goal state after:
A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
a : A
y : ℝ≥0
ha : 0 ≤ a
x : ℝ
hx : x ∈ quasispectrum ℝ a
⊢ (fun x ↦ ↑(x.toNNReal.nnrpow y)) x = (fun x ↦ x ^ ↑y) x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x y : ℝ≥0
hx : 0 < x
hy : 0 < y
⊢ a ^ (x + y) = a ^ x * a ^ y

Tactic:
by
  simp only [nnrpow_def]
  rw [← cfcₙ_mul _ _ a]
  congr! 2 with z
  exact mod_cast z.rpow_add' <| ne_of_gt (add_pos hx hy)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ ∀ x ∈ quasispectrum ℝ a, ‖x‖ ≤ max ‖a⁺‖ ‖a⁻‖

Tactic:
intro x hx

Goal state after:
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
hx : x ∈ quasispectrum ℝ a
⊢ ‖x‖ ≤ max ‖a⁺‖ ‖a⁻‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ {a | 0 ≤ a} ⊆ ↑(span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 r))

Tactic:
intro x hx

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
x : A
hx : x ∈ {a | 0 ≤ a}
⊢ x ∈ ↑(span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 r))
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«0»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)‖ ≤ ‖a‖

Tactic:
constructor
      · simp
        cfc_tac

Goal state after:
case h.left.«0».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«1»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)‖ ≤ ‖a‖

Tactic:
constructor
      · simp
        cfc_tac

Goal state after:
case h.left.«1».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«2»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)‖ ≤ ‖a‖

Tactic:
constructor
      · simp
        cfc_tac

Goal state after:
case h.left.«2».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«3»
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩) ∧
    ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)‖ ≤ ‖a‖

Tactic:
constructor
      · simp
        cfc_tac

Goal state after:
case h.left.«3».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)‖ ≤ ‖a‖
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
y : ℝ
ha : 0 ≤ a
⊢ Set.EqOn (fun x ↦ ↑(x.toNNReal ^ y)) (fun x ↦ x ^ y) (spectrum ℝ a)

Tactic:
intro x hx

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
y : ℝ
ha : 0 ≤ a
x : ℝ
hx : x ∈ spectrum ℝ a
⊢ (fun x ↦ ↑(x.toNNReal ^ y)) x = (fun x ↦ x ^ y) x
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 : Aˣ
ha : IsStrictlyPositive ↑a
⊢ ↑a ^ 0 = 1

Tactic:
exact rpow_zero _

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

Goal state before:
pp: A : Type u_2
inst✝⁹ : NormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedAlgebra ℝ A
inst✝⁶ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : ContinuousStar A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
this : Set.EqOn (cfc f) (cfc fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ spectrum ℝ≥0 a ⊆ s}
⊢ Set.MapsTo (Real.toNNReal ∘ NNReal.toReal) s s

Tactic:
intro x hx

Goal state after:
A : Type u_2
inst✝⁹ : NormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedAlgebra ℝ A
inst✝⁶ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : ContinuousStar A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
this : Set.EqOn (cfc f) (cfc fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ spectrum ℝ≥0 a ⊆ s}
x : ℝ≥0
hx : x ∈ s
⊢ (Real.toNNReal ∘ NNReal.toReal) x ∈ s
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : IsClosed {a | 0 ≤ a}
a : Unitization ℂ A
⊢ (∃ x, 0 ≤ x ∧ ↑x = a) ↔ ∃ x, 0 ≤ a ∧ ↑x = a

Tactic:
congr! 2 with x

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : IsClosed {a | 0 ≤ a}
a : Unitization ℂ A
x : A
⊢ 0 ≤ x ∧ ↑x = a ↔ 0 ≤ a ∧ ↑x = a
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝¹⁰ : NonUnitalCStarAlgebra A
inst✝⁹ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : Module ℂ E
inst✝⁶ : SMul A E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : Module ℂ F
inst✝³ : SMul A F
inst✝² : CStarModule A E
inst✝¹ : CStarModule A F
inst✝ : StarOrderedRing A
⊢ ∀ {z : ℂ} {x y : C⋆ᵐᵒᵈ(A, E × F)}, ⟪x.1, (z • y).1⟫_A + ⟪x.2, (z • y).2⟫_A = z • (⟪x.1, y.1⟫_A + ⟪x.2, y.2⟫_A)

Tactic:
simp [smul_add]

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
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ Set.EqOn (fun x ↦ 1 - (1 + x)⁻¹) (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) (σₙ ℝ a)

Tactic:
intro x hx

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
x : ℝ
hx : x ∈ σₙ ℝ a
⊢ (fun x ↦ 1 - (1 + x)⁻¹) x = (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) x
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ Set.EqOn (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) (fun x ↦ 1 - (1 + x)⁻¹) (σₙ ℝ b)

Tactic:
intro x hx

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
x : ℝ
hx : x ∈ σₙ ℝ b
⊢ (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) x = (fun x ↦ 1 - (1 + x)⁻¹) x
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case e_f
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
i : n
⊢ (Finset.univ.sup fun b ↦ ‖diagonal v i b‖₊) = ‖v i‖₊

Tactic:
refine le_antisymm (Finset.sup_le fun j hj => ?_) ?_

Goal state after:
case e_f.refine_1
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
i j : n
hj : j ∈ Finset.univ
⊢ ‖diagonal v i j‖₊ ≤ ‖v i‖₊

case e_f.refine_2
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
i : n
⊢ ‖v i‖₊ ≤ Finset.univ.sup fun b ↦ ‖diagonal v i b‖₊
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
s : Set Y
inst✝ : CompactSpace ↑s
f : X → Y → E
g : C(↑s, E)
f_ae_contOn : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) s
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ y ∈ s, ‖f x y‖ ≤ bound x
⊢ ∀ᵐ (x : X) ∂μ, ∀ (y : ↑s), ‖s.domRestrict (f x) y‖ ≤ bound x

Tactic:
· simpa

Goal state after:
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝⁴ : MeasurableSpace X
μ : Measure X
inst✝³ : TopologicalSpace Y
E : Type u_3
inst✝² : NormedAddCommGroup E
s : Set Y
inst✝¹ : CompactSpace ↑s
inst✝ : Zero ↑s
f : X → Y → E
g : C(↑s, E)₀
f_ae_contOn : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) s
f_ae_zero : ∀ᵐ (x : X) ∂μ, f x ↑0 = 0
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ y ∈ s, ‖f x y‖ ≤ bound x
⊢ ∀ᵐ (x : X) ∂μ, ∀ (y : ↑s), ‖s.domRestrict (f x) y‖ ≤ bound x

Tactic:
· simpa

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

Goal state before:
pp: A : Type u_2
inst✝¹¹ : NonUnitalNormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedSpace ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : ContinuousStar A
inst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this : Set.EqOn (cfcₙ f) (cfcₙ fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
⊢ Set.MapsTo (Real.toNNReal ∘ NNReal.toReal) s s

Tactic:
intro x hx

Goal state after:
A : Type u_2
inst✝¹¹ : NonUnitalNormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedSpace ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : ContinuousStar A
inst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this : Set.EqOn (cfcₙ f) (cfcₙ fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
x : ℝ≥0
hx : x ∈ s
⊢ (Real.toNNReal ∘ NNReal.toReal) x ∈ s
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Projection

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
x : A
⊢ IsIdempotentElem (star x * x) ↔ IsIdempotentElem (x * star x)

Tactic:
simp [isIdempotentElem_iff_quasispectrum_subset ℝ, quasispectrum.mul_comm]

Goal state after: