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

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))

Tactic:
obtain (rfl | hl) := l.eq_or_neBot

Goal state after:
case inl
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f ⊥ (spectrum R a)
hF : ∀ᶠ (x : X) in ⊥, ContinuousOn (F x) (spectrum R a)
⊢ Tendsto (fun x ↦ cfc (F x) a) ⊥ (𝓝 (cfc f a))

case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : CommSemiring R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
h_cont : Continuous[inst✝, topologicalSpace R R {spectrum R a}] fun x ↦ (ofFun {spectrum R a}) (f x)
hf : ∀ (x : X), ContinuousOn (f x) (spectrum R a)
⊢ Continuous[inst✝, inst✝³] fun x ↦ cfc (f x) a

Tactic:
by
  rw [← continuousOn_univ] at h_cont ⊢
  exact h_cont.cfc_fun (fun x _ ↦ hf x)

Goal state after:
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
this : a ^ (1 / 2) = a ^ ↑(1 / 2)
⊢ sqrt a = a ^ (1 / 2)

Tactic:
sqrt_eq_nnrpow a

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
this : a ^ (1 / 2) = a ^ ↑(1 / 2)
⊢ a ^ (1 / 2) = a ^ (1 / 2)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: case neg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ¬0 ≤ a
ha' : ¬0 ≤ ↑a
⊢ CFC.sqrt ↑a = ↑(CFC.sqrt a)

Tactic:
rw [CFC.sqrt, CFC.sqrt, cfcₙ_apply_of_not_predicate _ ha,
      cfcₙ_apply_of_not_predicate _ ha', inr_zero]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case neg.inr
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
ha : ¬IsStrictlyPositive a
ha' : ¬IsUnit (sqrt a)
H : ¬IsUnit a
⊢ sqrt a⁻¹ʳ = (sqrt a)⁻¹ʳ

Tactic:
simp [inverse_non_unit _ ha', inverse_non_unit _ H]

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
x : A
⊢ x ∈ Icc 0 1 ↔ 0 ≤ x ∧ ‖x‖ ≤ 1

Tactic:
by
  simpa only [map_one] using mem_Icc_algebraMap_iff_norm_le zero_le_one (A := A)

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 b : Aˣ
ha : 0 ≤ ↑a
hab : ↑a ≤ ↑b
hb : 0 ≤ ↑b
hb_inv : 0 ≤ ↑b⁻¹
ha_inv : 0 ≤ ↑a⁻¹
⊢ ↑b⁻¹ ≤ ↑a⁻¹

Tactic:
rw [le_iff_norm_sqrt_mul_sqrt_inv ha hb, ← sq_le_one_iff₀ (norm_nonneg _), sq,
    ← CStarRing.norm_star_mul_self] at hab

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : Aˣ
ha : 0 ≤ ↑a
hab : ‖star (sqrt ↑a * sqrt ↑b⁻¹) * (sqrt ↑a * sqrt ↑b⁻¹)‖ ≤ 1
hb : 0 ≤ ↑b
hb_inv : 0 ≤ ↑b⁻¹
ha_inv : 0 ≤ ↑a⁻¹
⊢ ↑b⁻¹ ≤ ↑a⁻¹
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 b : Aˣ
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
⊢ ↑a⁻¹ ≤ ↑b ↔ ↑b⁻¹ ≤ ↑a

Tactic:
simpa using CStarAlgebra.inv_le_inv_iff ha (inv_nonneg_of_nonneg b hb)

Goal state after:
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
a b : A
hb : IsSelfAdjoint b
this : ∀ (a b : Unitization ℂ A), IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a)
⊢ ↑(star a * b * a) ≤ ↑(‖b‖ • (star a * a))

Tactic:
simpa [Unitization.norm_inr] using this a b <| hb.inr ℂ

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 b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
⊢ ContinuousOn (fun x ↦ (1 + x)⁻¹) (σ ℝ≥0 c)

Tactic:
fun_prop (disch := intro _ _; positivity)

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

Goal state before:
pp: 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
hs : s ∈ 𝓝ˢ (⋃ x, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a Set.univ
ha' : ∀ (x : X), 0 ≤ a x
hf : ContinuousOn f s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ

Tactic:
exact ha_cont.cfc_nnreal_of_mem_nhdsSet f (by simpa) (by simpa)

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
⊢ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) a ≤ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b

Tactic:
calc _ = cfcₙ (fun x : ℝ≥0 => 1 - (1 + x)⁻¹) a := by
          rw [cfcₙ_nnreal_eq_real _ _ ha]
          refine cfcₙ_congr ?_
          intro x hx
          have hx' : 0 ≤ x := by grind
          simp [hx']
    _ ≤ cfcₙ (fun x : ℝ≥0 => 1 - (1 + x)⁻¹) b :=
          CFC.monotoneOn_one_sub_one_add_inv ha hb hab
    _ = cfcₙ (fun x : ℝ => 1 - (1 + x)⁻¹) b := by
          rw [cfcₙ_nnreal_eq_real _ _ hb]
          refine cfcₙ_congr ?_
          intro x hx
          have hx' : 0 ≤ x := by grind
          simp [hx']

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

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

Tactic:
obtain (rfl | hl) := l.eq_or_neBot

Goal state after:
case inl
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f ⊥ (quasispectrum R a)
hF : ∀ᶠ (x : X) in ⊥, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in ⊥, F x 0 = 0
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) ⊥ (𝓝 (cfcₙ f a))

case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

Tactic:
open scoped NonUnitalContinuousFunctionalCalculus in
  obtain (rfl | hl) := l.eq_or_neBot
  · simp
  have hf := h_tendsto.continuousOn hF.frequently
  have hf0 : f 0 = 0 := Eq.symm <|
    tendsto_nhds_unique (tendsto_const_nhds.congr' <| .symm hF0) <|
    h_tendsto.tendsto_at (quasispectrum.zero_mem R a)
  by_cases ha : p a
  · let s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
    have hs : s ∈ l := hF.and hF0
    rw [← tendsto_comap'_iff (i := ((↑) : s → X)) (by simpa)]
    conv =>
      enter [1, x]
      rw [Function.comp_apply, cfcₙ_apply (hf := x.2.1) (hf0 := x.2.2)]
    rw [cfcₙ_apply ..]
    apply cfcₙHom_continuous _ |>.tendsto _ |>.comp
    rw [ContinuousMapZero.isEmbedding_toContinuousMap.isInducing.tendsto_nhds_iff]
    change Tendsto (fun x : s ↦ (⟨_, x.2.1.domRestrict⟩ : C(quasispectrum R a, R))) _
      (𝓝 ⟨_, hf.domRestrict⟩)
    rw [hf.tendsto_domRestrict_iff_tendstoUniformlyOn (fun x ↦ x.2.1)]
    intro t
    simp only [eventually_comap, Subtype.forall]
    peel h_tendsto t with ht x _
    simp_all
  · simpa [cfcₙ_apply_of_not_predicate a ha] using tendsto_const_nhds

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
⊢ ContinuousOn f (g '' σₙ ℝ≥0 a)

Tactic:
fun_prop (disch := intro _ _; positivity)

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
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
⊢ ContinuousOn g Set.univ

Tactic:
fun_prop (disch := intro _ _; positivity)

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha : IsStarNormal a
⊢ quasispectrum ℝ ↑(ℑ a) = im '' quasispectrum ℂ a

Tactic:
by
  simp [← (ℑ a).2.quasispectrumRestricts.image, quasispectrum_imaginaryPart a, Set.image_image]

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
⊢ (fun x ↦ (ofMatrix.symm x.1, ofMatrix.symm x.2)) = id

Tactic:
ext i <;> rfl

Goal state after:
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
f : ℝ → ℝ
hf : f ∈ (fun x ↦ cfc x a) ⁻¹' {a | 0 ≤ a}
hf' : ContinuousOn f (spectrum ℝ a)
ha' : IsSelfAdjoint a
⊢ (fun x ↦ cfc x a) f = (cfcHom ha') { toFun := (spectrum ℝ a).domRestrict f, continuous_toFun := ⋯ }

Tactic:
by simp [cfc_apply f a ha' hf']

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
⊢ cfcₙ (fun z ↦ ∫ (x : X), f x z ∂μ) a = ∫ (x : X), cfcₙ (f x) a ∂μ

Tactic:
have key₂ (z : quasispectrum 𝕜 a) :
      ∫ x, f x z ∂μ = mkD ((quasispectrum 𝕜 a).domRestrict (fun z ↦ ∫ x, f x z ∂μ)) 0 z := by
    rw [mkD_apply_of_continuousOn]
    · rw [continuousOn_iff_continuous_domRestrict]
      refine continuous_congr key₁ |>.mpr ?_
      exact map_continuous (∫ x, mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ)
    · exact integral_eq_zero_of_ae hf₂

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
key₂ :
  ∀ (z : ↑(quasispectrum 𝕜 a)),
    ∫ (x : X), f x ↑z ∂μ = (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z
⊢ cfcₙ (fun z ↦ ∫ (x : X), f x z ∂μ) a = ∫ (x : X), cfcₙ (f x) a ∂μ
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: case refine_1.a
𝕜 : Type u_1
ι : Type u_4
M : ι → Type u_5
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : (i : ι) → Set (M i)
x : (i : ι) → M i
hx : ∀ (i : ι), x i ∈ s i
h :
  ∀ (x₁ : (i : ι) → M i),
    (∀ (i : ι), x₁ i ∈ s i) → ∀ (x₂ : (i : ι) → M i), (∀ (i : ι), x₂ i ∈ s i) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x
i : ι
x₁ : M i
hx₁ : x₁ ∈ s i
x₂ : M i
hx₂ : x₂ ∈ s i
hi : x i ∈ openSegment 𝕜 x₁ x₂
⊢ x ∈ openSegment 𝕜 (update x i x₁) (update x i x₂)

pp: 𝕜 : Type u_1
ι : Type u_4
M : ι → Type u_5
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : (i : ι) → Set (M i)
x : (i : ι) → M i
hx : ∀ (i : ι), x i ∈ s i
h :
  ∀ (x₁ : (i : ι) → M i),
    (∀ (i : ι), x₁ i ∈ s i) → ∀ (x₂ : (i : ι) → M i), (∀ (i : ι), x₂ i ∈ s i) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x
i : ι
x₁ : M i
hx₁ : x₁ ∈ s i
x₂ : M i
hx₂ : x₂ ∈ s i
hi : x i ∈ openSegment 𝕜 x₁ x₂
⊢ ∀ (i_1 : ι), update x i x₁ i_1 ∈ s i_1

Tactic:
· rw [← Pi.image_update_openSegment]
      exact ⟨_, hi, update_eq_self _ _⟩

Goal state after:
𝕜 : Type u_1
ι : Type u_4
M : ι → Type u_5
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : (i : ι) → Set (M i)
x : (i : ι) → M i
hx : ∀ (i : ι), x i ∈ s i
h :
  ∀ (x₁ : (i : ι) → M i),
    (∀ (i : ι), x₁ i ∈ s i) → ∀ (x₂ : (i : ι) → M i), (∀ (i : ι), x₂ i ∈ s i) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x
i : ι
x₁ : M i
hx₁ : x₁ ∈ s i
x₂ : M i
hx₂ : x₂ ∈ s i
hi : x i ∈ openSegment 𝕜 x₁ x₂
⊢ ∀ (i_1 : ι), update x i x₁ i_1 ∈ s i_1
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: h : exp Real.pi = 1
heq : cexp (↑Real.pi * I) = 1
⊢ False

Tactic:
exp_pi_mul_I

Goal state after:
h : exp Real.pi = 1
heq : -1 = 1
⊢ False
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Y'
eX eX' : OpenPartialHomeomorph X X'
eY eY' : OpenPartialHomeomorph Y Y'
h : (eX'.prod eY').source.Nonempty
⊢ eX = eX' ∧ eY' = eY ↔ eX = eX' ∧ eY = eY'

Tactic:
@eq_comm _ eY'

Goal state after:
X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Y'
eX eX' : OpenPartialHomeomorph X X'
eY eY' : OpenPartialHomeomorph Y Y'
h : (eX'.prod eY').source.Nonempty
⊢ eX = eX' ∧ eY = eY' ↔ eX = eX' ∧ eY = eY'
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_3
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
E : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (b : B) → TopologicalSpace (E b)
inst✝ : FiberBundle F E
s : (x : B) → E x
x₀ : B
⊢ ContinuousWithinAt (fun x ↦ TotalSpace.mk' F x (s x)) univ x₀ ↔
    ContinuousWithinAt (fun x ↦ (↑(trivializationAt F E x₀) { proj := x, snd := s x }).2) univ x₀

Tactic:
exact continuousWithinAt_section F

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

Goal state before:
pp: case h₂
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
⊢ e.baseSet = e'.baseSet

Tactic:
exact h₃

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

Goal state before:
pp: case fst
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : SMulCommClass 𝕜 A A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : StarRing 𝕜
inst✝² : StarRing A
inst✝¹ : StarModule 𝕜 A
inst✝ : NormedStarGroup A
x✝² : 𝕜
x✝¹ : 𝓜(𝕜, A)
x✝ : A
⊢ (star (x✝² • x✝¹)).toProd.1 x✝ = (star x✝² • star x✝¹).toProd.1 x✝

Tactic:
exact star_smul _ _

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

Goal state before:
pp: case snd
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : SMulCommClass 𝕜 A A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : StarRing 𝕜
inst✝² : StarRing A
inst✝¹ : StarModule 𝕜 A
inst✝ : NormedStarGroup A
x✝² : 𝕜
x✝¹ : 𝓜(𝕜, A)
x✝ : A
⊢ (star (x✝² • x✝¹)).toProd.2 x✝ = (star x✝² • star x✝¹).toProd.2 x✝

Tactic:
exact star_smul _ _

Goal state after:
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)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
h1 : ∀ (b : A), C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2
⊢ ∀ (x : A), ‖f x‖₊ ≤ ?m.208 * ?m.209 * ‖x‖₊

Tactic:
simpa only [sqrt_sq, sqrt_mul] using fun b ↦ sqrt_le_sqrt.2 <| (h b).trans (h1 b)

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

Goal state before:
pp: z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : |(↑z).arg| < π / ↑n
h2 : |(n • ↑(↑z).arg).toReal| < s
hs0 : 0 < s
hn0 : n ≠ 0
hn : 1 ≤ ↑n
⊢ ↑n * |(↑z).arg| < s

Tactic:
(Angle.nsmul_toReal_eq_mul hn0).mpr (mem_Ioc_of_Ioo ?_),

Goal state after:
z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : |(↑z).arg| < π / ↑n
h2 : |↑n * (↑(↑z).arg).toReal| < s
hs0 : 0 < s
hn0 : n ≠ 0
hn : 1 ≤ ↑n
⊢ ↑n * |(↑z).arg| < s

z : Circle
s : ℝ
n : ℕ
hs : s ≤ π
h1 : |(↑z).arg| < π / ↑n
h2 : |(n • ↑(↑z).arg).toReal| < s
hs0 : 0 < s
hn0 : n ≠ 0
hn : 1 ≤ ↑n
⊢ (↑(↑z).arg).toReal ∈ Ioo (-π / ↑n) (π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case inr
x y : Circle
hne : x ≠ y
⊢ (range ⇑(x.path y))ᶜ.Nonempty

Tactic:
compl_range_path hne

Goal state after:
case inr
x y : Circle
hne : x ≠ y
⊢ (⇑(y.path x) '' Ioo 0 1).Nonempty
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
hne : x ≠ y
⊢ 0 < ⟨2⁻¹, ⋯⟩

Tactic:
by simp [← unitInterval.coe_pos]

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
x : A
hx : x ≠ 0
u₁ : ↥(unitary A) := selfAdjoint.unitarySelfAddISMul (ℜ (‖x‖⁻¹ • x)) ⋯
⊢ ‖ℑ (‖x‖⁻¹ • x)‖ ≤ 1

Tactic:
simpa [norm_smul, inv_mul_le_one₀ (norm_pos_iff.2 hx)] using! imaginaryPart.norm_le x

Goal state after:
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
⊢ ContinuousAt (fun x ↦ cfc (fun x ↦ ↑x.arg) ↑x) u

Tactic:
have hε : 0 < ε := lt_of_le_of_lt (by positivity) huε

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
⊢ ContinuousAt (fun x ↦ cfc (fun x ↦ ↑x.arg) ↑x) u
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : ‖↑u - 1‖ < 2
⊢ Real.arccos (1 - ‖↑u - 1‖ ^ 2 / 2) < Real.arccos (1 - 2 ^ 2 / 2)

Tactic:
apply Real.arccos_lt_arccos (by norm_num) (by gcongr)
        linarith [(by positivity : 0 ≤ ‖(u - 1 : A)‖ ^ 2 / 2)]

Goal state after: