mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
⊢ σ 𝕜 x = σ 𝕜 ↑x ∪ ⋃ z ∈ σ 𝕜 x \ σ 𝕜 ↑x, connectedComponentIn (σ 𝕜 ↑x)ᶜ z

Tactic:
by
  suffices IsClopen ((σ 𝕜 (x : A))ᶜ ↓∩ (σ 𝕜 x \ σ 𝕜 (x : A))) by
    rw [← this.biUnion_connectedComponentIn (sdiff_subset_compl _ _),
      union_sdiff_cancel (spectrum.subset_subalgebra x)]
  have : CompleteSpace S := hS.completeSpace_coe
  have h_open : IsOpen (σ 𝕜 x \ σ 𝕜 (x : A)) := by
    rw [← (spectrum.isClosed (𝕜 := 𝕜) x).closure_eq, closure_eq_interior_union_frontier,
      union_sdiff_distrib, sdiff_eq_empty.mpr (frontier_spectrum S x),
      sdiff_eq_compl_inter, union_empty]
    exact (spectrum.isClosed _).isOpen_compl.inter isOpen_interior
  apply isClopen_preimage_val h_open
  suffices h_frontier : frontier (σ 𝕜 x \ σ 𝕜 (x : A)) ⊆ frontier (σ 𝕜 (x : A)) from
    disjoint_of_subset_left h_frontier <| disjoint_compl_right.frontier_left
      (spectrum.isClosed _).isOpen_compl
  grw [sdiff_eq_compl_inter, frontier_inter_subset, inter_subset_left, inter_subset_right,
    frontier_compl, frontier_subset_frontier, union_self]

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

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : QuasispectrumRestricts a ⇑Complex.reCLM
⊢ star a = cfcₙ id a

Tactic:
← cfcₙ_star_id a (R := ℂ)

Goal state after:
case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : QuasispectrumRestricts a ⇑Complex.reCLM
⊢ cfcₙ (fun x ↦ star x) a = cfcₙ id a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a b : A
ha : 0 ≤ a
hb : IsSelfAdjoint b ∧ QuasispectrumRestricts b ⇑ContinuousMap.realToNNReal
h : Commute a b
⊢ QuasispectrumRestricts (a * b) ⇑ContinuousMap.realToNNReal

Tactic:
obtain ⟨x, hx₁, hx₂, rfl⟩ := CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts hb.1 hb.2

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : PartialOrder A
inst✝⁷ : StarRing A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module ℝ A
inst✝³ : IsScalarTower ℝ A A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
x : A
hx₁ : IsSelfAdjoint x
hx₂ : QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal
hb : IsSelfAdjoint (x * x) ∧ QuasispectrumRestricts (x * x) ⇑ContinuousMap.realToNNReal
h : Commute a (x * x)
⊢ QuasispectrumRestricts (a * (x * x)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x✝¹ : E
x✝ : x✝¹ ∈ s
⊢ ‖(B x✝¹) 0‖ ≤ 1

Tactic:
simp only [map_zero, norm_zero, zero_le_one]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), approx n c x ≤ approx (n + 1) c x
c : CU P
⊢ approx (n + 1) c x ≤ approx (n + 1 + 1) c x

Tactic:
rw [approx, approx]
    exact midpoint_le_midpoint (ihn _) (ihn _)

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ dist (midpoint ℝ (c.left.lim a✝) 0) (midpoint ℝ (c.left.lim x) 0) ≤ 3 / 4 * (3 / 4) ^ n

Tactic:
refine (dist_midpoint_midpoint_le _ _ _ _).trans ?_

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n
c : CU P
hxl : x ∈ c.left.U
a✝ : X
hyl : a✝ ∈ c.left.U
hyd : dist (c.left.lim a✝) (c.left.lim x) ≤ (3 / 4) ^ n
⊢ (dist (c.left.lim a✝) (c.left.lim x) + dist 0 0) / 2 ≤ 3 / 4 * (3 / 4) ^ n
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case pos
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 : C(X, ℝ≥0)
hgI : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I
hg : ∀ (x : X), g x ≤ 1
hgt : Set.EqOn (⇑g) 1 t
x : X
hx : x ∈ t
⊢ ‖f x - f x * ((↑(algebraMapCLM ℝ≥0 𝕜)).comp g) x‖₊ < ε

Tactic:
simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapCLM_apply,
        map_one, mul_one, sub_self, nnnorm_zero] using! hε

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

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

Tactic:
obtain ⟨k, k_comp, k_closed, sk, kt⟩ : ∃ k, IsCompact k ∧ IsClosed k ∧ s ⊆ interior k ∧ k ⊆ tᶜ :=
    exists_compact_closed_between hs ht.isOpen_compl hd.symm.subset_compl_left
  let P : Set X → Set X → Prop := fun C _ => IsCompact C
  set c : Urysohns.CU P :=
  { C := k
    U := tᶜ
    P_C_U := k_comp
    closed_C := k_closed
    open_U := ht.isOpen_compl
    subset := kt
    hP := by
      rintro c u - c_comp u_open cu
      rcases exists_compact_closed_between c_comp u_open cu with ⟨k, k_comp, k_closed, ck, ku⟩
      have A : closure (interior k) ⊆ k :=
        (IsClosed.closure_subset_iff k_closed).2 interior_subset
      refine ⟨interior k, isOpen_interior, ck, A.trans ku, c_comp,
        k_comp.of_isClosed_subset isClosed_closure A⟩ }
  exact ⟨⟨c.lim, c.continuous_lim⟩, fun x hx ↦ c.lim_of_mem_C _ (sk.trans interior_subset hx),
    fun x hx => c.lim_of_notMem_U _ fun h => h hx, c.lim_mem_Icc⟩

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
inst✝¹ : T2Space X
inst✝ : RCLike 𝕜
x y : X
hxy : (continuousMapEval X 𝕜) x = (continuousMapEval X 𝕜) y
⊢ x = y

Tactic:
contrapose! hxy
    rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.IsClosed {x})
        (isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
      ⟨f, fx, fy, -⟩
    rw [DFunLike.ne_iff]
    use (⟨fun (x : ℝ) => (x : 𝕜), RCLike.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
    simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne,
      RCLike.ofReal_inj] using
      ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)

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

Goal state before:
pp: case neg
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : ¬f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)

Tactic:
simp only [cfcₙ_apply_of_not_map_zero _ hf₀, Pi.zero_def]

Goal state after:
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
𝔖 : Set (Set α)
inst✝ : PseudoEMetricSpace β
f : γ → α →ᵤ[𝔖] β
K : Set α → ℝ≥0
h : ∀ s ∈ 𝔖, ∀ c ∈ s, LipschitzWith (K s) fun x ↦ (toFun 𝔖) (f x) c
s : Set α
hs : s ∈ 𝔖
y : α
hy : y ∈ s
⊢ LipschitzWith (K s) fun x ↦ UniformFun.toFun ((⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(toFun 𝔖) ∘ f) x) ⟨y, hy⟩

Tactic:
exact h s hs y hy

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : CStarRing A
inst✝⁵ : NormedSpace 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 b : A
ha : 0 ≤ a
⊢ ‖star b * a * b‖ = ‖sqrt a * b‖ * ‖sqrt a * b‖

Tactic:
← CStarRing.norm_star_mul_self,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : CStarRing A
inst✝⁵ : NormedSpace 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 b : A
ha : 0 ≤ a
⊢ ‖star b * a * b‖ = ‖star (sqrt a * b) * (sqrt a * b)‖
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
ha : IsSelfAdjoint a
hb : IsSelfAdjoint b
⊢ (∀ x ∈ spectrum ℝ ↑(b - a), 0 ≤ x) ↔ 0 ≤ b - a

Tactic:
← Unitization.quasispectrum_eq_spectrum_inr' ℝ ℂ

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
ha : IsSelfAdjoint a
hb : IsSelfAdjoint b
⊢ (∀ x ∈ σₙ ℝ (b - a), 0 ≤ x) ↔ 0 ≤ b - a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : Aˣ
hb : IsStrictlyPositive ↑b
hbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))
| ‖sqrt a * ↑b ^ (-(1 / 2))‖ * ‖sqrt a * ↑b ^ (-(1 / 2))‖ ≤ 1

Tactic:
← CStarRing.norm_star_mul_self,

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
b : Aˣ
hb : IsStrictlyPositive ↑b
hbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))
| ‖star (sqrt a * ↑b ^ (-(1 / 2))) * (sqrt a * ↑b ^ (-(1 / 2)))‖ ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.Module.Defs

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℂ A
inst✝⁶ : Module ℂ E
inst✝⁵ : PartialOrder A
inst✝⁴ : SMul A E
inst✝³ : Norm A
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarModule ℂ A
z : ℝ
x y : E
h₁ : z •> x = ↑z •> x
⊢ star ⟪y, ↑z •> x⟫ = z •> ⟪x, y⟫

Tactic:
inner_smul_right_complex

Goal state after:
A : Type u_1
E : Type u_2
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℂ A
inst✝⁶ : Module ℂ E
inst✝⁵ : PartialOrder A
inst✝⁴ : SMul A E
inst✝³ : Norm A
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarModule ℂ A
z : ℝ
x y : E
h₁ : z •> x = ↑z •> x
⊢ star (↑z •> ⟪y, x⟫) = z •> ⟪x, y⟫
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 : A
ha : 0 ≤ a
x : ℂ
hx : x ∈ σₙ ℂ a
⊢ 0 ≤ x

Tactic:
Unitization.quasispectrum_eq_spectrum_inr' ℂ ℂ a

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
x : ℂ
hx : x ∈ spectrum ℂ ↑a
⊢ 0 ≤ 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 : A
ha : 1 ≤ a
ha' : 0 ≤ a
n : ℕ
⊢ ∀ ⦃b : ℕ⦄, n ≤ b → (fun x ↦ a ^ x) n ≤ (fun x ↦ a ^ x) b

Tactic:
m

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 1 ≤ a
ha' : 0 ≤ a
n m : ℕ
⊢ n ≤ m → (fun x ↦ a ^ x) n ≤ (fun x ↦ a ^ x) m
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
n : ℕ
⊢ ∀ ⦃b : ℕ⦄, n ≤ b → (fun x ↦ a ^ x) b ≤ (fun x ↦ a ^ x) n

Tactic:
m

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
n m : ℕ
⊢ n ≤ m → (fun x ↦ a ^ x) m ≤ (fun x ↦ a ^ x) n
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) b = cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b

Tactic:
rw [cfcₙ_nnreal_eq_real _ _ hb]

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
⊢ cfcₙ (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) b = cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
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
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := ⋯
hs : s ∈ l
⊢ Tendsto (fun x ↦ { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
    (comap Subtype.val l) (𝓝 { toFun := (quasispectrum R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })

Tactic:
rw [ContinuousMapZero.isEmbedding_toContinuousMap.isInducing.tendsto_nhds_iff]

Goal state after:
case pos
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
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto
    (toContinuousMap ∘ fun x ↦
      { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
    (comap Subtype.val l) (𝓝 ↑{ toFun := (quasispectrum R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hN : ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
hxn : ‖x‖₊ = 1
⊢ ∑ j, ‖A i j‖₊ ≤ N

Tactic:
rw [hxn, mul_one, Pi.nnnorm_def, Finset.sup_le_iff] at hN

Goal state after:
case inr
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hN : ∀ b ∈ Finset.univ, ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x b‖₊ ≤ N
hxn : ‖x‖₊ = 1
⊢ ∑ j, ‖A i j‖₊ ≤ N
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
C : ℝ≥0
hmain : ∀ (a : A₁), 0 ≤ a → ‖f a‖ ≤ ↑C * ‖a‖
x : A₁
y : Fin 4 → A₁
hy_nonneg : ∀ (i : Fin 4), 0 ≤ y i
hy_norm : ∀ (i : Fin 4), ‖y i‖ ≤ ‖x‖
hy : x = ∑ i, I ^ ↑i • y i
⊢ ∑ i, ↑C * ‖x‖ ≤ ↑(4 * C) * ‖x‖

Tactic:
by simp [mul_assoc]

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

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A₁
inst✝⁵ : NonUnitalCStarAlgebra A₂
inst✝⁴ : PartialOrder A₁
inst✝³ : PartialOrder A₂
inst✝² : StarOrderedRing A₁
inst✝¹ : StarOrderedRing A₂
n : Type u_3
inst✝ : Fintype n
φ : A₁ →CP A₂
M : CStarMatrix n n A₁
hM : 0 ≤ M
k : ℕ := Fintype.card n
e : n ≃ Fin (Fintype.card n) := Fintype.equivFinOfCardEq ⋯
hmain : 0 ≤ (reindexₐ ℂ A₂ e) ((mapₗ ↑φ) M)
⊢ 0 ≤ M.map ⇑φ

Tactic:
simpa [reindexₐ_symm] using map_nonneg (reindexₐ ℂ A₂ e).symm hmain

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

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
S : Type u_4
A : Type u_5
B : Type u_6
inst✝² : Fintype m
inst✝¹ : NonUnitalCStarAlgebra A
inst✝ : Fintype n
⊢ ∀ (x y : CStarMatrix n n A),
    (↑(MulOpposite.opLinearEquiv ℂ) ∘ₗ toCLM).toFun (x * y) =
      (↑(MulOpposite.opLinearEquiv ℂ) ∘ₗ toCLM).toFun x * (↑(MulOpposite.opLinearEquiv ℂ) ∘ₗ toCLM).toFun y

Tactic:
intros
      simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, LinearMap.coe_comp,
        LinearEquiv.coe_coe, MulOpposite.coe_opLinearEquiv, Function.comp_apply,
        ← MulOpposite.op_mul, MulOpposite.op_inj]
      ext
      simp [toCLM]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖M i j‖ * ‖M i j‖ = 0

Tactic:
← CStarRing.norm_star_mul_self,

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖star (M i j) * M i j‖ = 0
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 i j‖ ≤ ‖M‖ * ‖M i j‖

Tactic:
← CStarRing.norm_star_mul_self,

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
⊢ ‖star (M i j) * M i j‖ ≤ ‖M‖ * ‖M i j‖
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
M : CStarMatrix m n A
⊢ ∑ x, ∑ x, ‖ofMatrixₗ.symm M‖ = ↑(↑(Fintype.card n) * ↑(Fintype.card m)) * ‖ofMatrixₗ.symm M‖

Tactic:
by simp [mul_assoc]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ star (cfc f a) * cfc f a = 1 ∧ cfc f a * star (cfc f a) = 1 ↔ ∀ x ∈ spectrum R a, star (f x) * f x = 1

Tactic:
rw [← IsStarNormal.cfc_map (p := p) f a |>.star_comm_self |>.eq, and_self, ← cfc_one R a,
    ← cfc_star, ← cfc_mul .., cfc_eq_cfc_iff_eqOn]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ Set.EqOn (fun x ↦ star (f x) * f x) 1 (spectrum R a) ↔ ∀ x ∈ spectrum R a, star (f x) * f x = 1
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A B C : Set E
hAB : IsExtreme 𝕜 A B
hAC : IsExtreme 𝕜 A C
⊢ IsExtreme 𝕜 A (B ∩ C)

Tactic:
use Subset.trans inter_subset_left hAB.1
  rintro x₁ hx₁A x₂ hx₂A x ⟨hxB, hxC⟩ hx
  exact ⟨hAB.left_mem_of_mem_openSegment hx₁A hx₂A hxB hx,
    hAC.left_mem_of_mem_openSegment hx₁A hx₂A hxC hx⟩

Goal state after:
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
j : ι
⊢ update x i x₂ j ∈ s j

Tactic:
obtain rfl | hji := eq_or_ne j i

Goal state after:
case refine_1.a.inl
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
j : ι
x₁ : M j
hx₁ : x₁ ∈ s j
x₂ : M j
hx₂ : x₂ ∈ s j
hi : x j ∈ openSegment 𝕜 x₁ x₂
⊢ update x j x₂ j ∈ s j

case refine_1.a.inr
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
j : ι
hji : j ≠ i
⊢ update x i x₂ j ∈ s j