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

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict ((s • 1) • f), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (s • 1) • (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }

Tactic:
rw [← map_smul]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict ((s • 1) • f), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) ((s • 1) • { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ ¬star (f 0) = 0

Tactic:
exact fun hf0 ↦ h0 <| by simpa using congr(star $(hf0))

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
ν n✝ : ℕ
⊢ Polynomial.derivative (bernsteinPolynomial R (n✝ + 1) (ν + 1)) =
    ↑(n✝ + 1) * (bernsteinPolynomial R (n✝ + 1 - 1) ν - bernsteinPolynomial R (n✝ + 1 - 1) (ν + 1))

Tactic:
rw [Nat.cast_succ]

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
ν n✝ : ℕ
⊢ Polynomial.derivative (bernsteinPolynomial R (n✝ + 1) (ν + 1)) =
    (↑n✝ + 1) * (bernsteinPolynomial R (n✝ + 1 - 1) ν - bernsteinPolynomial R (n✝ + 1 - 1) (ν + 1))
Module: Mathlib.Topology.ContinuousMap.Polynomial

Goal state before:
pp: case refine_2
R : Type u_1
α : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
g : R[X]
f : C(α, R)
x : α
n : ℕ
⊢ ∀ (a : R), ((aeval f) ((monomial n) a)) x = eval (f x) ((monomial n) a)

Tactic:
a

Goal state after:
case refine_2
R : Type u_1
α : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
g : R[X]
f : C(α, R)
x : α
n : ℕ
a : R
⊢ ((aeval f) ((monomial n) a)) x = eval (f x) ((monomial n) a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case e'_3
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
n : ℕ
k : 𝕜
hk : k ∈ σ a
pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))
nnnorm_pow_le : ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊
hn : 0 < ↑(n + 1)
⊢ ↑‖k‖₊ = (↑‖k‖₊ ^ (n + 1)) ^ (1 / ↑(n + 1))

Tactic:
one_div,

Goal state after:
case e'_3
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
n : ℕ
k : 𝕜
hk : k ∈ σ a
pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))
nnnorm_pow_le : ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊
hn : 0 < ↑(n + 1)
⊢ ↑‖k‖₊ = (↑‖k‖₊ ^ (n + 1)) ^ (↑(n + 1))⁻¹
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
n : ℕ
⊢ spectralRadius 𝕜 a ≤ ↑‖a ^ (n + 1)‖₊ ^ (1 / (↑n + 1)) * ↑‖1‖₊ ^ (1 / (↑n + 1))

Tactic:
refine iSup₂_le fun k hk => ?_
  -- apply easy direction of the spectral mapping theorem for polynomials
  have pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1)) := by
    simpa only [one_mul, Algebra.algebraMap_eq_smul_one, one_smul, aeval_monomial, one_mul,
      eval_monomial] using subset_polynomial_aeval a (@monomial 𝕜 _ (n + 1) (1 : 𝕜)) ⟨k, hk, rfl⟩
  -- power of the norm is bounded by norm of the power
  have nnnorm_pow_le : (↑(‖k‖₊ ^ (n + 1)) : ℝ≥0∞) ≤ ‖a ^ (n + 1)‖₊ * ‖(1 : A)‖₊ := by
    simpa only [Real.toNNReal_mul (norm_nonneg _), norm_toNNReal, nnnorm_pow k (n + 1),
      ENNReal.coe_mul] using coe_mono (Real.toNNReal_mono (norm_le_norm_mul_of_mem pow_mem))
  -- take (n + 1)ᵗʰ roots and clean up the left-hand side
  have hn : 0 < ((n + 1 : ℕ) : ℝ) := mod_cast Nat.succ_pos'
  convert monotone_rpow_of_nonneg (one_div_pos.mpr hn).le nnnorm_pow_le
  all_goals dsimp
  · rw [one_div, pow_rpow_inv_natCast]
    positivity
  rw [Nat.cast_succ, ENNReal.coe_mul_rpow]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht✝ : TotallyBounded t
U : SetRel α α
hU : U ∈ 𝓤 α
u : Set α
hu : u.Finite
ht : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}
s : Set α
hs : s ⊆ t
x : α
hx : x ∈ (s, u ∩ U.image s).1
y : α
hy : y ∈ u
hxy : x ∈ {x | (x, y) ∈ U}
⊢ x ∈ U.preimage (s, u ∩ U.image s).2

Tactic:
exact ⟨y, ⟨hy, ⟨x, hx, hxy⟩⟩, hxy⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
⊢ ∀ᶠ (i : ℕ) in Filter.cofinite, ‖(↑(i + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ i‖ ≤ ‖a - ↑ₐ z‖ ^ i / ↑i.factorial

Tactic:
filter_upwards [Filter.eventually_cofinite_ne 0] with n hn

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
n : ℕ
hn : n ≠ 0
⊢ ‖(↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n‖ ≤ ‖a - ↑ₐ z‖ ^ n / ↑n.factorial
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:
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)]

Goal state after:
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
⊢ IsClopen ((σ 𝕜 ↑x)ᶜ ↓∩ σ 𝕜 x \ σ 𝕜 ↑x)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case refine_2.h.h
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, ‖↑t - x‖₊ ≤ t
x : ℝ
hx : x ∈ spectrum ℝ a
h_le : ‖↑t - x‖ ≤ ↑t
⊢ 0 ≤ x

Tactic:
Real.norm_eq_abs,

Goal state after:
case refine_2.h.h
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, ‖↑t - x‖₊ ≤ t
x : ℝ
hx : x ∈ spectrum ℝ a
h_le : |↑t - x| ≤ ↑t
⊢ 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra ℝ A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ f '' σ ℝ≥0 a = (fun x ↦ ‖↑(f x.toNNReal)‖₊) '' σ ℝ a

Tactic:
simp [Set.image_image, ← ha'.image]

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z ↦ resolvent a z
H₃ : resolvent a 0 = 0
⊢ False

Tactic:
exact not_isUnit_zero <| H₃ ▸ (isUnit_resolvent.mp <| H₀.symm ▸ Set.mem_univ 0)

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
ha' : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ f '' σₙ ℝ≥0 a = (fun x ↦ ‖↑(f x.toNNReal)‖₊) '' σₙ ℝ a

Tactic:
simp [Set.image_image, ← ha'.image]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
⊢ z ∈ ofReal ∘ re '' σ ℂ a → z ∈ σ ℂ a

Tactic:
by
    rintro ⟨z, hz, rfl⟩
    simpa only [(ha.mem_spectrum_eq_re hz).symm, Function.comp_apply] using hz

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E →ₗ[𝕜] 𝕜
r : ℝ
⊢ f.toSeminorm.ball 0 r = {x | ‖f x‖ < r}

Tactic:
simp only [Seminorm.ball_zero_eq, toSeminorm_apply]

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
⊢ GaloisConnection setOfIdeal (idealOfSet R)

Tactic:
refine fun I s => ⟨fun h f hf => ?_, fun h x hx => ?_⟩
  · by_contra h'
    rcases notMem_idealOfSet.mp h' with ⟨x, hx, hfx⟩
    exact hfx (notMem_setOfIdeal.mp (mt (@h x) hx) hf)
  · obtain ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx
    by_contra hx'
    exact notMem_idealOfSet.mpr ⟨x, hx', hfx⟩ (h hf)

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⁻

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
⊢ (-a)⁺ = a⁻

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

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
⊢ (-a)⁺ = a⁻
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
⊢ (-a)⁺ = a⁻

Tactic:
have ha' : ¬ IsSelfAdjoint (-a) := fun h ↦ ha (by simpa using h.neg)

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)
⊢ (-a)⁺ = a⁻
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
⊢ (-a)⁺ = a⁻

Tactic:
have ha' : ¬ IsSelfAdjoint (-a) := fun h ↦ ha (by simpa using h.neg)
    rw [posPart_def, negPart_def, cfcₙ_apply_of_not_predicate a ha,
      cfcₙ_apply_of_not_predicate _ ha']

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

Goal state before:
pp: 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
⊢ a⁺ - a⁺ = 0

Tactic:
sub_self

Goal state after:
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
⊢ 0 = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case refine_1
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)
⊢ IsCompact (σₙ ℝ a)

Tactic:
exact isCompact_quasispectrum _

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

Goal state before:
pp: case refine_2
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)
⊢ IsCompact (σₙ ℝ b)

Tactic:
exact isCompact_quasispectrum _

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

Goal state before:
pp: case refine_3
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)
⊢ IsCompact (σₙ ℝ (-c))

Tactic:
exact isCompact_quasispectrum _

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

Goal state before:
pp: case mpr.mem
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝¹ : PartialOrder A := spectralOrder A
x✝ p x : A
hx : x ∈ Set.range fun s ↦ star s * s
⊢ IsSelfAdjoint x ∧ QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal

Tactic:
obtain ⟨s, rfl⟩ := hx

Goal state after:
case mpr.mem
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p s : A
⊢ IsSelfAdjoint ((fun s ↦ star s * s) s) ∧ QuasispectrumRestricts ((fun s ↦ star s * s) s) ⇑ContinuousMap.realToNNReal
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 : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ cfc (fun x_1 ↦ x_1 ^ (-x)) ↑a = cfc (fun x_1 ↦ x_1 ^ x) (cfc (fun x ↦ x⁻¹) ↑a)

Tactic:
← cfc_comp' (fun z => z ^ x) (Inv.inv : ℝ≥0 → ℝ≥0) (a : A) h₁

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 : Aˣ
x : ℝ
ha' : 0 ≤ ↑a
h₁ : ContinuousOn (fun z ↦ z ^ x) (Inv.inv '' spectrum ℝ≥0 ↑a)
⊢ cfc (fun x_1 ↦ x_1 ^ (-x)) ↑a = cfc (fun x_1 ↦ x_1⁻¹ ^ x) ↑a
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
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConvexOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
⊢ ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y

Tactic:
a

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConvexOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a : ℝ
⊢ ∀ ⦃b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f 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
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConcaveOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
⊢ ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)

Tactic:
a

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConcaveOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a : ℝ
⊢ ∀ ⦃b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)
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 : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
ha_cont : ContinuousOn a Set.univ
ha : ∀ (x : X), spectrum ℝ≥0 (a x) ⊆ s
hf : ContinuousOn f s
ha' : ∀ (x : X), 0 ≤ a x
⊢ ContinuousOn (fun x ↦ cfc f (a x)) Set.univ

Tactic:
exact ha_cont.cfc_nnreal' hs f (fun x _ ↦ ha x) (fun x _ ↦ ha' x)

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

Goal state before:
pp: n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
⊢ (Finset.univ.sup fun b ↦ Finset.univ.sup fun b_1 ↦ ‖diagonal v b b_1‖₊) = Finset.univ.sup fun b ↦ ‖v b‖₊

Tactic:
congr 1 with i : 1

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

Goal state before:
pp: m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖diagonal v i j‖₊) = Finset.univ.sup fun b ↦ ‖v b‖₊

Tactic:
congr 1 with i : 1

Goal state after:
case e_f
m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
i : m
⊢ ∑ j, ‖diagonal v i j‖₊ = ‖v i‖₊
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✝ : RCLike α
A : Matrix l m α
B : Matrix m n α
⊢ (∑ i, ∑ j, ‖∑ j_1, A i j_1 * B j_1 j‖₊ ^ 2) ^ (1 / 2) ≤
    (∑ i, ∑ j, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ i, ∑ j, ‖B i j‖₊ ^ 2) ^ (1 / 2)

Tactic:
rw [← NNReal.mul_rpow, @Finset.sum_comm _ _ m, Finset.sum_mul_sum]

Goal state after:
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✝ : RCLike α
A : Matrix l m α
B : Matrix m n α
⊢ (∑ i, ∑ j, ‖∑ j_1, A i j_1 * B j_1 j‖₊ ^ 2) ^ (1 / 2) ≤ (∑ i, ∑ j, (∑ j, ‖A i j‖₊ ^ 2) * ∑ x, ‖B x j‖₊ ^ 2) ^ (1 / 2)
Module: Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap

Goal state before:
pp: F : Type u_1
A₁ : Type u_2
A₂ : Type u_3
inst✝⁷ : NonUnitalCStarAlgebra A₁
inst✝⁶ : NonUnitalCStarAlgebra A₂
inst✝⁵ : PartialOrder A₁
inst✝⁴ : PartialOrder A₂
inst✝³ : StarOrderedRing A₁
inst✝² : StarOrderedRing A₂
inst✝¹ : FunLike F A₁ A₂
inst✝ : LinearMapClass F ℂ A₁ A₂
h : ∀ (φ : F) (k : ℕ) (M : CStarMatrix (Fin k) (Fin k) A₁), 0 ≤ M → 0 ≤ M.map ⇑φ
φ : F
⊢ ∀ (x : A₁), 0 ≤ x → 0 ≤ φ x

Tactic:
a

Goal state after:
F : Type u_1
A₁ : Type u_2
A₂ : Type u_3
inst✝⁷ : NonUnitalCStarAlgebra A₁
inst✝⁶ : NonUnitalCStarAlgebra A₂
inst✝⁵ : PartialOrder A₁
inst✝⁴ : PartialOrder A₂
inst✝³ : StarOrderedRing A₁
inst✝² : StarOrderedRing A₂
inst✝¹ : FunLike F A₁ A₂
inst✝ : LinearMapClass F ℂ A₁ A₂
h : ∀ (φ : F) (k : ℕ) (M : CStarMatrix (Fin k) (Fin k) A₁), 0 ≤ M → 0 ≤ M.map ⇑φ
φ : F
a : A₁
⊢ 0 ≤ a → 0 ≤ φ a
Module: Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : TopologicalSpace Y
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompactSpace Y
f : X → C(Y, E)
bound : X → ℝ
bound_int : HasFiniteIntegral bound μ
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ bound x
h : Nonempty Y
⊢ HasFiniteIntegral f μ

Tactic:
have bound_nonneg : 0 ≤ᵐ[μ] bound := by
      filter_upwards [bound_ge] with x bound_x using le_trans (norm_nonneg _) (bound_x h.some)
    refine .mono' bound_int ?_
    filter_upwards [bound_ge, bound_nonneg] with x bound_ge_x bound_nonneg_x
    exact ContinuousMap.norm_le _ bound_nonneg_x |>.mpr bound_ge_x

Goal state after: