mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖⁻¹ * dist p₁ p₂

Tactic:
midpoint_comm,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ : P
⊢ dist (midpoint 𝕜 p₂ p₁) p₂ = ‖2‖⁻¹ * dist p₁ p₂
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: case h.right
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
⊢ ∀ ⦃x y : WeakDual 𝕜 E⦄, x ≠ y → ∃ n, x (denseSeq E n) ≠ y (denseSeq E n)

Tactic:
intro w

Goal state after:
case h.right
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
w : WeakDual 𝕜 E
⊢ ∀ ⦃y : WeakDual 𝕜 E⦄, w ≠ y → ∃ n, w (denseSeq E n) ≠ y (denseSeq E n)
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
h : 0 < r * (b - a)
⊢ f ((lineMap a b) r) ≤ (lineMap (f a) (f b)) r ↔ slope f a ((lineMap a b) r) ≤ slope f a b

Tactic:
by
  rw [lineMap_apply, lineMap_apply, slope, slope, vsub_eq_sub, vsub_eq_sub, vsub_eq_sub,
    vadd_eq_add, vadd_eq_add, smul_eq_mul, add_sub_cancel_right, smul_sub, smul_sub, smul_sub,
    sub_le_iff_le_add, mul_inv_rev, mul_smul, mul_smul, ← smul_sub, ← smul_sub, ← smul_add,
    smul_smul, ← mul_inv_rev, inv_smul_le_iff_of_pos h, smul_smul,
    mul_inv_cancel_right₀ (right_ne_zero_of_mul h.ne'), smul_add,
    smul_inv_smul₀ (left_ne_zero_of_mul h.ne')]

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 t ∧ EqOn (⇑f) 1 s ∧ ∀ (x : X), f x ∈ Icc 0 1

Tactic:
by
  obtain ⟨g, hgs, hgt, (hicc : ∀ x, 0 ≤ g x ∧ g x ≤ 1)⟩ := exists_continuous_zero_one_of_isCompact
    hs ht hd
  use 1 - g
  refine ⟨?_, ?_, ?_⟩
  · intro x hx
    simp only [ContinuousMap.sub_apply, ContinuousMap.one_apply, Pi.zero_apply]
    exact sub_eq_zero_of_eq (hgt.symm hx)
  · intro x hx
    simp only [ContinuousMap.sub_apply, ContinuousMap.one_apply, Pi.one_apply, sub_eq_self]
    exact hgs hx
  · intro x
    simpa [and_comm] using hicc x

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

Goal state before:
pp: case pos
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
⊢ (-a)⁺ = a⁻

Tactic:
rw [posPart_def, negPart_def, ← cfcₙ_comp_neg _ _]
    congr! 2

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
⊢ a⁺ = cfcₙ id a

Tactic:
posPart_def

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
⊢ cfcₙ (fun x ↦ x⁺) a = cfcₙ id a
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
h✝ : Nonempty α
this : BddAbove (Set.range fun x ↦ dist (toFun f x) (toFun g x))
⊢ edist f g = ENNReal.ofReal (⨆ x, dist (toFun f x) (toFun g x))

Tactic:
exact ENNReal.eq_of_forall_le_nnreal_iff fun r ↦ by simp [edist_def, ciSup_le_iff this]

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

Goal state before:
pp: case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Surjective (characterSpaceToSpectrum a)

Tactic:
rintro ⟨z, hz⟩

Goal state after:
case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
z : ℂ
hz : z ∈ spectrum ℂ a
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩
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

Tactic:
← sub_eq_zero

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 mpr.zero
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p : A
⊢ IsSelfAdjoint 0 ∧ QuasispectrumRestricts 0 ⇑ContinuousMap.realToNNReal

Tactic:
rw [quasispectrumRestricts_iff_spectrumRestricts_inr' ℂ, SpectrumRestricts.nnreal_iff]
          simp

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

Goal state before:
pp: case mpr
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p : A
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ IsSelfAdjoint p ∧ QuasispectrumRestricts p ⇑ContinuousMap.realToNNReal

Tactic:
induction hp using AddSubmonoid.closure_induction with
        | mem x hx =>
          obtain ⟨s, rfl⟩ := hx
          refine ⟨IsSelfAdjoint.star_mul_self s, ?_⟩
          rw [quasispectrumRestricts_iff_spectrumRestricts_inr' ℂ,
            SpectrumRestricts.nnreal_iff, Unitization.inr_mul, Unitization.inr_star]
          exact spectrum_star_mul_self_nonneg
        | zero =>
          rw [quasispectrumRestricts_iff_spectrumRestricts_inr' ℂ, SpectrumRestricts.nnreal_iff]
          simp
        | add x y _ _ hx hy =>
          simp +singlePass only [← Unitization.isSelfAdjoint_inr (R := ℂ),
            quasispectrumRestricts_iff_spectrumRestricts_inr' ℂ] at hx hy ⊢
          rw [Unitization.inr_add]
          exact ⟨hx.1.add hy.1, hx.2.nnreal_add hx.1 hy.1 hy.2⟩

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
⊢ ‖b * a * star b‖ = ‖b * sqrt a‖ ^ 2

Tactic:
conv_rhs => rw [← (CFC.sqrt_nonneg a).star_eq, ← star_star b, ← star_mul, norm_star,
    ← norm_star_mul_mul_self_of_nonneg _ ha, star_star]

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

Goal state before:
pp: 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
⊢ ↑(cfcₙ f a) = cfc f ↑a

Tactic:
by_cases h : ContinuousOn f (σₙ R a) ∧ p a

Goal state after:
case pos
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
⊢ ↑(cfcₙ f a) = cfc f ↑a

case neg
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)
⊢ ↑(cfcₙ f a) = cfc f ↑a
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
⊢ cfcₙ (fun x ↦ ↑(x.toNNReal.nnrpow y)) a = cfcₙ (fun x ↦ x ^ ↑y) a

Tactic:
refine cfcₙ_congr ?_

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
⊢ Set.EqOn (fun x ↦ ↑(x.toNNReal.nnrpow y)) (fun x ↦ x ^ ↑y) (quasispectrum ℝ a)
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
ha : 0 ≤ a
⊢ a ^ 1 = a

Tactic:
by
  simp only [nnrpow_def, NNReal.nnrpow_def, NNReal.coe_one, NNReal.rpow_one]
  change cfcₙ (id : ℝ≥0 → ℝ≥0) a = a
  rw [cfcₙ_id ℝ≥0 a]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case inr
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}
hx_pos : 0 < ‖x‖
⊢ x ∈ ↑(span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 r))

Tactic:
suffices (r * ‖x‖⁻¹ : ℂ)⁻¹ • ((r * ‖x‖⁻¹ : ℂ) • x) = x by
      rw [← this]
      refine smul_mem _ _ (subset_span <| Set.mem_inter ?_ ?_)
      · norm_cast
        exact smul_nonneg (by positivity) hx
      · simp [mul_smul, norm_smul, abs_of_pos hr, inv_mul_cancel₀ hx_pos.ne']
    apply inv_smul_smul₀
    norm_cast
    positivity

Goal state after:
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 : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
⊢ sqrt a ^ x = a ^ (x / 2)

Tactic:
rw [sqrt_eq_nnrpow, nnrpow_nnrpow, one_div_mul_eq_div 2 x]

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✝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
s : Set X
h_cont : ContinuousOn (fun x ↦ (ofFun {spectrum R a}) (f x)) s
hf : ∀ x ∈ s, ContinuousOn (f x) (spectrum R a)
⊢ ContinuousOn (fun x ↦ cfc (f x) a) s

Tactic:
ContinuousOn

Goal state after:
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
s : Set X
h_cont : ∀ x ∈ s, ContinuousWithinAt (fun x ↦ (ofFun {spectrum R a}) (f x)) s x
hf : ∀ x ∈ s, ContinuousOn (f x) (spectrum R a)
⊢ ∀ x ∈ s, ContinuousWithinAt (fun x ↦ cfc (f x) a) s x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
y : ℝ
hy : y ≠ 0
ha : 0 ≤ a
a✝ : Nontrivial A
h : ∀ x ∈ spectrum ℝ≥0 a, x ^ y ≠ 0
hf : ContinuousOn (fun x ↦ x ^ y) (spectrum ℝ≥0 a)
⊢ 0 ∉ spectrum ℝ≥0 a

Tactic:
intro h0

Goal state after:
case pos
A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
y : ℝ
hy : y ≠ 0
ha : 0 ≤ a
a✝ : Nontrivial A
h : ∀ x ∈ spectrum ℝ≥0 a, x ^ y ≠ 0
hf : ContinuousOn (fun x ↦ x ^ y) (spectrum ℝ≥0 a)
h0 : 0 ∈ spectrum ℝ≥0 a
⊢ False
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
⊢ cfc (⇑NNReal.sqrt) ((algebraMap ℝ≥0 A) r) = (algebraMap ℝ≥0 A) (NNReal.sqrt r)

Tactic:
cfc_algebraMap

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
⊢ (algebraMap ℝ≥0 A) (NNReal.sqrt r) = (algebraMap ℝ≥0 A) (NNReal.sqrt r)
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:
rw [sqrt_eq_cfc, cfc_algebraMap]

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✝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
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t

Tactic:
ContinuousOn

Goal state after:
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
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ∀ x ∈ t, ContinuousWithinAt (fun x ↦ cfc f (a x)) t x
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 : 0 ≤ a
hab : a ≤ b
⊢ ∀ (a b : Unitization ℂ A), 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖

Tactic:
intro a b ha hab

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ b✝ : A
ha✝ : 0 ≤ a✝
hab✝ : a✝ ≤ b✝
a b : Unitization ℂ A
ha : 0 ≤ a
hab : a ≤ b
⊢ ‖a‖ ≤ ‖b‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: case pos
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
inrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A
hf₀ : f 0 = 0
h₁ : inr ∘ cfcₙ f = fun x ↦ ↑(cfcₙ f x)
⊢ ConvexOn ℝ s (inr ∘ cfcₙ f)

Tactic:
have h₂ : (fun x : A => ((cfcₙ f x : A) : A⁺¹))
      = fun x : A => cfc f (x : A⁺¹) := by ext1; rw [real_cfcₙ_eq_cfc_inr ..]

Goal state after:
case pos
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
inrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A
hf₀ : f 0 = 0
h₁ : inr ∘ cfcₙ f = fun x ↦ ↑(cfcₙ f x)
h₂ : (fun x ↦ ↑(cfcₙ f x)) = fun x ↦ cfc f ↑x
⊢ ConvexOn ℝ s (inr ∘ cfcₙ f)
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
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t

Tactic:
ContinuousOn

Goal state after:
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
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum ℝ≥0 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ∀ x ∈ t, ContinuousWithinAt (fun x ↦ cfc f (a x)) t 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 b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
h_cfc_one_sub :
  ∀ (c : Unitization ℂ A),
    autoParam (0 ≤ c) monotoneOn_one_sub_one_add_inv._auto_1_6 →
      cfc (fun x ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)
⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b

Tactic:
h_cfc_one_sub (a : A⁺¹),

Goal state after:
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
h_cfc_one_sub :
  ∀ (c : Unitization ℂ A),
    autoParam (0 ≤ c) monotoneOn_one_sub_one_add_inv._auto_1_6 →
      cfc (fun x ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (fun x ↦ x⁻¹) (1 + c)
⊢ 1 - cfc (fun x ↦ x⁻¹) (1 + ↑a) ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b
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.toNNReal)⁻¹)) a

Tactic:
refine cfcₙ_congr ?_

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
⊢ Set.EqOn (fun x ↦ 1 - (1 + x)⁻¹) (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) (σₙ ℝ a)
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.toNNReal)⁻¹)) b = cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b

Tactic:
refine cfcₙ_congr ?_

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
⊢ Set.EqOn (fun x ↦ ↑(1 - (1 + x.toNNReal)⁻¹)) (fun x ↦ 1 - (1 + x)⁻¹) (σₙ ℝ b)
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✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : SMulCommClass R A A
inst✝² : IsScalarTower R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
s : Set X
h_cont : ContinuousOn (fun x ↦ (ofFun {quasispectrum R a}) (f x)) s
hf : ∀ x ∈ s, ContinuousOn (f x) (quasispectrum R a)
hf0 : ∀ x ∈ s, f x 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ (f x) a) s

Tactic:
ContinuousOn

Goal state after:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : SMulCommClass R A A
inst✝² : IsScalarTower R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
s : Set X
h_cont : ∀ x ∈ s, ContinuousWithinAt (fun x ↦ (ofFun {quasispectrum R a}) (f x)) s x
hf : ∀ x ∈ s, ContinuousOn (f x) (quasispectrum R a)
hf0 : ∀ x ∈ s, f x 0 = 0
⊢ ∀ x ∈ s, ContinuousWithinAt (fun x ↦ cfcₙ (f x) a) s x
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ ‖replicateCol ι v‖₊ = ‖v‖₊

Tactic:
by
  simp [nnnorm_def, Pi.nnnorm_def]

Goal state after: