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

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
s : Set α
x✝ : s ∈ {t | t ⊆ L ∧ ∀ L ∈ M '' u, (t ∩ L).Nonempty}
hsL : s ⊆ L
hsu : ∀ ⦃x : Set α⦄, x ∈ u → (s ∩ M x).Nonempty
⊢ ∃ y ∈ SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty}, s ⤳ y

Tactic:
let s' : Compacts α := ⟨L ∩ closure s, hL.inter_right isClosed_closure⟩

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
s : Set α
x✝ : s ∈ {t | t ⊆ L ∧ ∀ L ∈ M '' u, (t ∩ L).Nonempty}
hsL : s ⊆ L
hsu : ∀ ⦃x : Set α⦄, x ∈ u → (s ∩ M x).Nonempty
s' : Compacts α := { carrier := L ∩ closure[inst✝³] s, isCompact' := ⋯ }
⊢ ∃ y ∈ SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty}, s ⤳ y
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹² : AddCommGroup E
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : IsRCLikeNormedField 𝕜
inst✝⁹ : TopologicalSpace E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : PolynormableSpace 𝕜 E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space F
S : Submodule 𝕜 E
f : ↥S →L[𝕜] F
inst✝ : FiniteDimensional 𝕜 ↥(↑f).range
⊢ ∃ g, f = g ∘SL S.subtypeL

Tactic:
by
  let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
  let b := Module.finBasis 𝕜 f.range
  let e := b.equivFunL
  let fi := fun i ↦ (LinearMap.toContinuousLinearMap (b.coord i)).comp
    (f.codRestrict _ <| LinearMap.mem_range_self _)
  choose gi hgf using fun i ↦ StrongDual.exists_extension S (fi i)
  use f.range.subtypeL.comp <| e.symm.toContinuousLinearMap.comp (.pi gi)
  ext x
  simp [fi, e, hgf]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
c : CU P
n : ℕ
x : X
⊢ approx n c x ∈ Icc (approx n c.right x) (approx n c.left x)

Tactic:
induction n generalizing c with
  | zero =>
    simp only [approx]
    refine ⟨le_rfl, ?_⟩
    grw [left_U_subset]
    rw [Pi.one_apply]; positivity -- TODO: `positivity` doesn't prove that `1 x` is nonnegative
  | succ n ihn =>
    simp only [approx, mem_Icc]
    refine ⟨midpoint_le_midpoint ?_ (ihn _).1, midpoint_le_midpoint (ihn _).2 ?_⟩ <;>
      apply approx_le_approx_of_U_sub_C
    exacts [subset_closure, subset_closure]

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
g : C(X, ℝ)
hgs : EqOn (⇑g) 0 s
hgt : EqOn (⇑g) 1 t
hicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1
⊢ ∃ f, EqOn (⇑f) 0 t ∧ EqOn (⇑f) 1 s ∧ ∀ (x : X), f x ∈ Icc 0 1

Tactic:
use 1 - g

Goal state after:
case h
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)) 0 t ∧ EqOn (⇑(1 - g)) 1 s ∧ ∀ (x : X), (1 - g) x ∈ Icc 0 1
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
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
⊢ ⇑{ toFun := g, continuous_toFun := ⋯ } ⁻¹' {1} ⊆ s

Tactic:
apply compl_subset_compl.1

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 := ⋯
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
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
ha' : ¬IsSelfAdjoint (-a)
⊢ cfcₙ (fun x ↦ x⁺) (-a) = cfcₙ (fun x ↦ x⁻) a

Tactic:
cfcₙ_apply_of_not_predicate a ha,

Goal state after:
case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
ha' : ¬IsSelfAdjoint (-a)
⊢ cfcₙ (fun x ↦ x⁺) (-a) = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
this : Fact (0 ∈ s)
mul₁ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hb' hbs) f * (cfcₙHomSuperset hc' hcs) g = 0
mul₂ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hc' hcs) f * (cfcₙHomSuperset hb' hbs) g = 0
ψ : C(↑s, ℝ)₀ →⋆ₙₐ[ℝ] A :=
  let __src := ↑(cfcₙHomSuperset hb' hbs) + ↑(cfcₙHomSuperset hc' hcs);
  { toFun := ⇑(cfcₙHomSuperset hb' hbs) + ⇑(cfcₙHomSuperset hc' hcs), map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯,
    map_mul' := ⋯, map_star' := ⋯ }
key : cfcₙHomSuperset ha has = ψ
f : C(↑s, ℝ)₀ := { toFun := fun x ↦ (↑x)⁺, continuous_toFun := ⋯, map_zero' := ⋯ }
⊢ a⁺ = b

Tactic:
replace key := congr($key f)

Goal state after:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
this : Fact (0 ∈ s)
mul₁ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hb' hbs) f * (cfcₙHomSuperset hc' hcs) g = 0
mul₂ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hc' hcs) f * (cfcₙHomSuperset hb' hbs) g = 0
ψ : C(↑s, ℝ)₀ →⋆ₙₐ[ℝ] A :=
  let __src := ↑(cfcₙHomSuperset hb' hbs) + ↑(cfcₙHomSuperset hc' hcs);
  { toFun := ⇑(cfcₙHomSuperset hb' hbs) + ⇑(cfcₙHomSuperset hc' hcs), map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯,
    map_mul' := ⋯, map_star' := ⋯ }
f : C(↑s, ℝ)₀ := { toFun := fun x ↦ (↑x)⁺, continuous_toFun := ⋯, map_zero' := ⋯ }
key : (cfcₙHomSuperset ha has) f = ψ f
⊢ a⁺ = b
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ ↑(x - y) = ↑0

Tactic:
rw [Unitization.inr_zero, Unitization.inr_sub]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ ↑x - ↑y = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case neg.inr
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
ha : ¬p a
⊢ ↑(cfcₙ f a) = cfc f ↑a

Tactic:
cfcₙ_apply_of_not_predicate a ha,

Goal state after:
case neg.inr
A : Type u_1
inst✝¹³ : NonUnitalCStarAlgebra A
R : Type u_2
inst✝¹² : Semifield R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : Algebra R ℂ
inst✝³ : IsScalarTower R ℂ A
p : A → Prop
p' : Unitization ℂ A → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R A p
inst✝¹ : ContinuousFunctionalCalculus R (Unitization ℂ A) p'
inst✝ : ContinuousMapZero.UniqueHom R (Unitization ℂ A)
hp : ∀ {a : A}, p' ↑a ↔ p a
a : A
f : R → R
hf₀ : f 0 = 0
h : ¬(ContinuousOn f (σₙ R a) ∧ p a)
ha : ¬p a
⊢ ↑0 = cfc f ↑a
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: 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‖

pp: 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‖

Tactic:
· exact CStarAlgebra.norm_negPart_le _ |>.trans <| realPart.norm_le a

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
r : ℝ≥0
⊢ sqrt ((algebraMap ℝ≥0 A) r) = (algebraMap ℝ≥0 A) (NNReal.sqrt r)

Tactic:
sqrt_eq_cfc,

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
r : ℝ≥0
⊢ cfc (⇑NNReal.sqrt) ((algebraMap ℝ≥0 A) r) = (algebraMap ℝ≥0 A) (NNReal.sqrt r)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case 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
x : ℝ≥0
htriv : 0 ≤ a
h₁ : 0 < x
h₂ : ↑x / 2 = ↑(x / 2)
h₃ : 0 < x / 2
⊢ sqrt (a ^ ↑x) = a ^ (↑x / 2)

Tactic:
← nnrpow_eq_rpow h₁,

Goal state after:
case 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
x : ℝ≥0
htriv : 0 ≤ a
h₁ : 0 < x
h₂ : ↑x / 2 = ↑(x / 2)
h₃ : 0 < x / 2
⊢ sqrt (a ^ x) = a ^ (↑x / 2)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum 𝕜 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), p (a x)
x : X
x✝ : x ∈ Set.univ
⊢ ∀ᶠ (x_1 : X) in 𝓝[Set.univ] x, spectrum 𝕜 (a x_1) ⊆ s x

Tactic:
by simpa using ha x

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
⊢ AntitoneOn Ring.inverse {a | IsStrictlyPositive a}

Tactic:
intro a (apos : IsStrictlyPositive a) b (bpos : IsStrictlyPositive b) hab

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
apos : IsStrictlyPositive a
b : A
bpos : IsStrictlyPositive b
hab : a ≤ b
⊢ Ring.inverse b ≤ Ring.inverse 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
x : C⋆ᵐᵒᵈ(A, E × F)
h : ⟪x.1, x.1⟫_A + ⟪x.2, x.2⟫_A = 0
⊢ x = 0

Tactic:
apply equiv A (E × F) |>.injective

Goal state after:
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
x : C⋆ᵐᵒᵈ(A, E × F)
h : ⟪x.1, x.1⟫_A + ⟪x.2, x.2⟫_A = 0
⊢ (equiv A (E × F)) x = (equiv A (E × F)) 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
ha' : ∀ (x : X), 0 ≤ a x
x : X
x✝ : x ∈ Set.univ
⊢ ∀ᶠ (x_1 : X) in 𝓝[Set.univ] x, spectrum ℝ≥0 (a x_1) ⊆ s x

Tactic:
by simpa using ha x

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

Goal state before:
pp: A : Type u_1
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℂ E
⊢ ∀ {x y z : E}, ⟪x, y + z⟫_ℂ = ⟪x, y⟫_ℂ + ⟪x, z⟫_ℂ

Tactic:
simp [_root_.inner_add_right]

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

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalSeminormedRing α
A : Matrix l m α
B : Matrix m n α
⊢ ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊

Tactic:
by
  simp_rw [linfty_opNNNorm_def, Matrix.mul_apply]
  calc
    (Finset.univ.sup fun i => ∑ k, ‖∑ j, A i j * B j k‖₊) ≤
        Finset.univ.sup fun i => ∑ k, ∑ j, ‖A i j‖₊ * ‖B j k‖₊ :=
      Finset.sup_mono_fun fun i _hi =>
        Finset.sum_le_sum fun k _hk => nnnorm_sum_le_of_le _ fun j _hj => nnnorm_mul_le _ _
    _ = Finset.univ.sup fun i => ∑ j, ‖A i j‖₊ * ∑ k, ‖B j k‖₊ := by
      simp_rw [@Finset.sum_comm m, Finset.mul_sum]
    _ ≤ Finset.univ.sup fun i => ∑ j, ‖A i j‖₊ * Finset.univ.sup fun i => ∑ j, ‖B i j‖₊ := by
      refine Finset.sup_mono_fun fun i _hi => ?_
      gcongr with j hj
      exact Finset.le_sup (f := fun i ↦ ∑ k : n, ‖B i k‖₊) hj
    _ ≤ (Finset.univ.sup fun i => ∑ j, ‖A i j‖₊) * Finset.univ.sup fun i => ∑ j, ‖B i j‖₊ := by
      simp_rw [← Finset.sum_mul, ← NNReal.finset_sup_mul]
      rfl

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
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 : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, quasispectrum 𝕜 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
hf0 : f 0 = 0
ha' : ∀ (x : X), p (a x)
x : X
x✝ : x ∈ Set.univ
⊢ ∀ᶠ (x_1 : X) in 𝓝[Set.univ] x, quasispectrum 𝕜 (a x_1) ⊆ s x

Tactic:
by simpa using ha x

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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : TopologicalSpace X
s : X → Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
hs : ∀ (x : X), IsCompact (s x)
ha : ∀ (x₀ : X), ∀ᶠ (x : X) in 𝓝 x₀, quasispectrum ℝ≥0 (a x) ⊆ s x₀
hf : ∀ (x : X), ContinuousOn f (s x)
hf0 : f 0 = 0
ha' : ∀ (x : X), 0 ≤ a x
x : X
x✝ : x ∈ Set.univ
⊢ ∀ᶠ (x_1 : X) in 𝓝[Set.univ] x, quasispectrum ℝ≥0 (a x_1) ⊆ s x

Tactic:
by simpa using ha x

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp [inner_smul_left]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
inner' : E →+ E →+ 𝕜 := { toFun := fun x ↦ ((innerₛₗ 𝕜) x).toAddMonoidHom, map_zero' := ⋯, map_add' := ⋯ }
this : Continuous fun p ↦ (inner' p.1) p.2
⊢ Continuous (uncurry (inner 𝕜))

Tactic:
Completion.toInner,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
inner' : E →+ E →+ 𝕜 := { toFun := fun x ↦ ((innerₛₗ 𝕜) x).toAddMonoidHom, map_zero' := ⋯, map_add' := ⋯ }
this : Continuous fun p ↦ (inner' p.1) p.2
⊢ Continuous (uncurry (inner 𝕜))
Module: Mathlib.Analysis.CStarAlgebra.Hom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CStarAlgebra A
inst✝³ : CStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
a : A
ha : IsSelfAdjoint a
φ : F
hφ : Function.Injective ⇑φ
h_spec : spectrum ℝ ((StarAlgHom.restrictScalars ℝ ↑φ) a) ⊆ spectrum ℝ a
x : ℝ
hx : x ∈ spectrum ℝ a
hx' : x ∉ spectrum ℝ (φ a)
f : C(ℝ, ℝ)
h_eqOn : Set.EqOn (⇑f) 0 (spectrum ℝ (φ a))
h_eqOn_x : Set.EqOn (⇑f) 1 {x}
this : cfc (⇑f) a = 0
⊢ False

Tactic:
← cfc_zero ℝ a,

Goal state after:
F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CStarAlgebra A
inst✝³ : CStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
a : A
ha : IsSelfAdjoint a
φ : F
hφ : Function.Injective ⇑φ
h_spec : spectrum ℝ ((StarAlgHom.restrictScalars ℝ ↑φ) a) ⊆ spectrum ℝ a
x : ℝ
hx : x ∈ spectrum ℝ a
hx' : x ∉ spectrum ℝ (φ a)
f : C(ℝ, ℝ)
h_eqOn : Set.EqOn (⇑f) 0 (spectrum ℝ (φ a))
h_eqOn_x : Set.EqOn (⇑f) 1 {x}
this : cfc (⇑f) a = cfc 0 a
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
⊢ ‖U‖ ≤ 1

Tactic:
simp_rw [pi_norm_le_iff_of_nonneg zero_le_one]

Goal state after:
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
⊢ ∀ (i i_1 : n), ‖U i i_1‖ ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:
pp: 𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m n 𝕜
⊢ ‖toContinuousLinearMap
        ((toLin (EuclideanSpace.basisFun n 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis) (Aᴴ * A))‖ =
    ‖A‖ * ‖A‖

Tactic:
Matrix.toLin_mul (v₂ := (EuclideanSpace.basisFun m 𝕜).toBasis),

Goal state after:
𝕜 : Type u_1
m : Type u_2
n : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m n 𝕜
⊢ ‖toContinuousLinearMap
        ((toLin (EuclideanSpace.basisFun m 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis) Aᴴ ∘ₗ
          (toLin (EuclideanSpace.basisFun n 𝕜).toBasis (EuclideanSpace.basisFun m 𝕜).toBasis) A)‖ =
    ‖A‖ * ‖A‖
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:

Tactic:
_ = ‖T (toLp 2 (Pi.single i (1 : 𝕜)))‖ := by
          rw [toEuclideanCLM_toLp (diagonal v) (Pi.single i (1 : 𝕜))]
          simp
      _ ≤ _ := by grw [T.le_opNorm]; simp

Goal state after: