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

Goal state before:
pp: case neg.inr.inl
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
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬f 0 = 0
⊢ P (cfcₙ f a)

pp: case neg.inr.inr
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
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)

Tactic:
· rwa [cfcₙ_apply_of_not_map_zero _ h]

Goal state after:
case neg.inr.inr
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
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝² : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this✝¹ : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
this✝ : ∀ᶠ (b : ℕ → α) in Filter.pi fun i ↦ 𝓝 (u i), ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2
J : Set ℕ
J_fin : J.Finite
k : ℕ → Set α
k_mem : ∀ (i : ℕ), k i ∈ 𝓝 (u i)
hk : J.pi k ⊆ {x | (fun b ↦ ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2) x}
v : ℕ → α
hv : ∀ i ∈ Iic n, f (v i) ∈ t i ∧ v i ∈ k i
h'v : ∀ i ∈ Iic n, v i ∈ s
this : c < ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i))
⊢ ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i)) ≤ eVariationOn f s

Tactic:
have v_mono : StrictMonoOn v (Iic n) := by
    let w i := if i ∈ Iic n then v i else u i
    suffices w ∈ J.pi k by
      simp only [Finset.mem_filter, Finset.mem_product, Finset.mem_Iic, and_imp, Prod.forall] at hk
      grind [StrictMonoOn]
    have A i : u i ∈ k i := mem_of_mem_nhds (k_mem i)
    grind

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝² : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this✝¹ : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
this✝ : ∀ᶠ (b : ℕ → α) in Filter.pi fun i ↦ 𝓝 (u i), ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2
J : Set ℕ
J_fin : J.Finite
k : ℕ → Set α
k_mem : ∀ (i : ℕ), k i ∈ 𝓝 (u i)
hk : J.pi k ⊆ {x | (fun b ↦ ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2) x}
v : ℕ → α
hv : ∀ i ∈ Iic n, f (v i) ∈ t i ∧ v i ∈ k i
h'v : ∀ i ∈ Iic n, v i ∈ s
this : c < ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i))
v_mono : StrictMonoOn v (Iic n)
⊢ ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i)) ≤ eVariationOn f s
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 0 (bernsteinPolynomial R n ν) = if ν = 0 then 1 else 0

Tactic:
rw [bernsteinPolynomial]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 0 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = 0 then 1 else 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 1 (bernsteinPolynomial R n ν) = if ν = n then 1 else 0

Tactic:
rw [bernsteinPolynomial]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 1 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = n then 1 else 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

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

Tactic:
rw [bernsteinPolynomial]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.derivative (↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) * (1 - X) ^ (n + 1 - (ν + 1))) =
    (↑n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1))
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ.succ
R : Type u_1
inst✝ : CommRing R
k : ℕ
ih : ∀ (n ν : ℕ), k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0
n ν : ℕ
⊢ k + 1 ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k + 1] (bernsteinPolynomial R n (ν + 1))) = 0

Tactic:
simp only [derivative_succ, Function.comp_apply,
        Function.iterate_succ, Polynomial.iterate_derivative_sub,
        Polynomial.iterate_derivative_natCast_mul, Polynomial.eval_mul, Polynomial.eval_natCast,
        Polynomial.eval_sub]
      intro h
      apply mul_eq_zero_of_right
      rw [ih _ _ (Nat.le_of_succ_le h), sub_zero]
      convert! ih _ _ (Nat.pred_le_pred h)
      exact (Nat.succ_pred_eq_of_pos (k.succ_pos.trans_le h)).symm

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

Goal state before:
pp: case mp.refine_2.convert_2
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (↑x + -a) * (b - a)⁻¹ ∈ I

Tactic:
← sub_eq_add_neg

Goal state after:
case mp.refine_2.convert_2
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
x : ↑(Set.Icc a b)
⊢ (↑x - a) * (b - a)⁻¹ ∈ I
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ (k + 1) • bernsteinPolynomial R n (k + 1) =
    ↑(k + 1) * X ^ (k + 1 - 1) * (1 - X) ^ (n - (k + 1)) * ↑(n.choose (k + 1)) * X

Tactic:
rw [bernsteinPolynomial]

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ (k + 1) • (↑(n.choose (k + 1)) * X ^ (k + 1) * (1 - X) ^ (n - (k + 1))) =
    ↑(k + 1) * X ^ (k + 1 - 1) * (1 - X) ^ (n - (k + 1)) * ↑(n.choose (k + 1)) * X
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
hf_neg : ContinuousOn (fun x ↦ -f x) (σₙ R a)
⊢ ContinuousOn f (σₙ R a)

Tactic:
simpa using hf_neg.fun_neg

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
n : ℕ
hn : n ≠ 0
f : C(↑I, E)
⊢ (bernsteinApproximation n f) 1 = f 1

Tactic:
simp [apply, Fin.sum_univ_castSucc, bernstein_apply, hn, Nat.sub_eq_zero_iff_le]

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

Goal state before:
pp: case succ.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ ((k + 1 + 1) * (k + 1 + 1 - 1)) • bernsteinPolynomial R n (k + 1 + 1) =
    ↑(n.choose (k + 1 + 1)) *
        ((1 - X) ^ (n - (k + 1 + 1)) * (↑(k + 1 + 1) * (↑(k + 1 + 1 - 1) * X ^ (k + 1 + 1 - 1 - 1)))) *
      X ^ 2

Tactic:
rw [bernsteinPolynomial]

Goal state after:
case succ.succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ ((k + 1 + 1) * (k + 1 + 1 - 1)) • (↑(n.choose (k + 1 + 1)) * X ^ (k + 1 + 1) * (1 - X) ^ (n - (k + 1 + 1))) =
    ↑(n.choose (k + 1 + 1)) *
        ((1 - X) ^ (n - (k + 1 + 1)) * (↑(k + 1 + 1) * (↑(k + 1 + 1 - 1) * X ^ (k + 1 + 1 - 1 - 1)))) *
      X ^ 2
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
⊢ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ⊆ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
intro f hf
  induction hf using adjoin_induction with
  | mem f hf =>
    obtain rfl := Set.mem_singleton_iff.mp hf
    rfl
  | add f g _ _ hf hg => exact add_mem hf hg
  | zero => exact zero_mem _
  | mul f g _ _ _ hg => exact Ideal.mul_mem_left _ f hg
  | smul r f _ hf =>
    rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
    rw [map_smul, hf, smul_zero]
  | star f _ hf =>
    rw [SetLike.mem_coe, RingHom.mem_ker] at hf ⊢
    rw [map_star, hf, star_zero]

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

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
⊢ φ (cfc f a) = cfc f (φ a)

Tactic:
have h_spec := AlgHom.spectrum_apply_subset ψ a

Goal state after:
F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
⊢ φ (cfc f a) = cfc f (φ a)
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
⊢ x ∈ U.preimage (s, u ∩ U.image s).2

Tactic:
obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht (hs hx))

Goal state after:
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
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)
⊢ exp z ∈ spectrum 𝕜 (exp a)

Tactic:
let b := ∑' n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ^ n

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
⊢ exp z ∈ spectrum 𝕜 (exp a)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝¹ : UniformSpace α
inst✝ : T0Space α
⊢ IsClosed (SetLike.coe ⁻¹' SetLike.coe '' Set.range fun x ↦ {x})

Tactic:
← Set.range_comp

Goal state after:
α : Type u_1
inst✝¹ : UniformSpace α
inst✝ : T0Space α
⊢ IsClosed (SetLike.coe ⁻¹' Set.range (SetLike.coe ∘ fun x ↦ {x}))
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
x✝ : CompactSpace (Closeds α)
ι : Type u_1
F : ι → Set α
hF₁ : ∀ (i : ι), IsClosed[inst✝.toTopologicalSpace] (F i)
hF₂ : ⋂ i, F i = ∅
this : {x | ∀ (i : ι), ↑x ⊆ F i} ⊆ {⊥} → ∃ u, {x | ∀ i ∈ u, ↑x ⊆ F i} ⊆ {⊥}
⊢ ∃ u, ⋂ i ∈ u, F i = ∅

Tactic:
← Set.subset_iInter_iff,

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
x✝ : CompactSpace (Closeds α)
ι : Type u_1
F : ι → Set α
hF₁ : ∀ (i : ι), IsClosed[inst✝.toTopologicalSpace] (F i)
hF₂ : ⋂ i, F i = ∅
this : {x | ↑x ⊆ ⋂ i, F i} ⊆ {⊥} → ∃ u, {x | ↑x ⊆ ⋂ i ∈ u, F i} ⊆ {⊥}
⊢ ∃ u, ⋂ i ∈ u, F i = ∅
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
⊢ K ∈ (fun y ↦ (y, L)) ⁻¹' Prod.map SetLike.coe SetLike.coe ⁻¹' hausdorffEntourage (id U)

Tactic:
Set.mem_preimage,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
⊢ Prod.map SetLike.coe SetLike.coe (K, L) ∈ hausdorffEntourage (id U)
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
l : Filter ↥S
a : ↥S
ha : IsUnit ↑a
hla : l ≤ 𝓝 a
hl : ∀ᶠ (x : ↥S) in l, IsUnit x
hl' : l.NeBot
⊢ Tendsto Ring.inverse (map (⇑(val S)) l) (𝓝 ↑ha.unit⁻¹)

Tactic:
rw [← Ring.inverse_unit]

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
l : Filter ↥S
a : ↥S
ha : IsUnit ↑a
hla : l ≤ 𝓝 a
hl : ∀ᶠ (x : ↥S) in l, IsUnit x
hl' : l.NeBot
⊢ Tendsto Ring.inverse (map (⇑(val S)) l) (𝓝 (Ring.inverse ↑ha.unit))
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
l : Filter ↥S
a : ↥S
ha : IsUnit ↑a
hla : l ≤ 𝓝 a
hl : ∀ᶠ (x : ↥S) in l, IsUnit x
hl' : l.NeBot
hla₂ : Tendsto Ring.inverse (map (⇑(val S)) l) (𝓝 ↑ha.unit⁻¹)
⊢ IsUnit a

Tactic:
suffices mem : (↑ha.unit⁻¹ : A) ∈ S by
    refine ⟨⟨a, ⟨(↑ha.unit⁻¹ : A), mem⟩, ?_, ?_⟩, rfl⟩
    all_goals ext; simp

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
l : Filter ↥S
a : ↥S
ha : IsUnit ↑a
hla : l ≤ 𝓝 a
hl : ∀ᶠ (x : ↥S) in l, IsUnit x
hl' : l.NeBot
hla₂ : Tendsto Ring.inverse (map (⇑(val S)) l) (𝓝 ↑ha.unit⁻¹)
⊢ ↑ha.unit⁻¹ ∈ S
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ∀ (x_1 : ↑(σ 𝕜 a)), ‖{ toFun := (σ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } x_1‖ ≤ ‖f x‖

Tactic:
rintro ⟨y, hy⟩

Goal state after:
case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : Nontrivial A
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σ 𝕜 a)
ha : p a
x : 𝕜
hx' : x ∈ σ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σ 𝕜 a) ((fun x ↦ ‖f x‖) x)
y : 𝕜
hy : y ∈ σ 𝕜 a
⊢ ‖{ toFun := (σ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } ⟨y, hy⟩‖ ≤ ‖f x‖
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
this : CompleteSpace ↥S
⊢ IsClopen ((σ 𝕜 ↑x)ᶜ ↓∩ σ 𝕜 x \ σ 𝕜 ↑x)

Tactic:
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

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
this : CompleteSpace ↥S
h_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 x \ σ 𝕜 ↑x)
⊢ IsClopen ((σ 𝕜 ↑x)ᶜ ↓∩ σ 𝕜 x \ σ 𝕜 ↑x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
⊢ ∀ (x_1 : ↑(σₙ 𝕜 a)), ‖↑{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } x_1‖ ≤ ‖f x‖

Tactic:
rintro ⟨y, hy⟩

Goal state after:
case e'_4.a
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun x ↦ ‖f x‖) x)
y : 𝕜
hy : y ∈ σₙ 𝕜 a
⊢ ‖↑{ toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } ⟨y, hy⟩‖ ≤ ‖f x‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖cfcₙ f a‖ ≤ c

Tactic:
by
  refine cfcₙ_cases (‖·‖ ≤ c) a f ?_ fun hf hf0 ha ↦ ?_
  · simpa using (norm_nonneg _).trans <| h 0 (quasispectrum.zero_mem 𝕜 a)
  · simp only [← cfcₙ_apply f a, isLUB_le_iff (IsGreatest.norm_cfcₙ f a hf hf0 ha |>.isLUB)]
    rintro - ⟨x, hx, rfl⟩
    exact h x hx

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

Goal state before:
pp: A : Type u_4
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
a : A
ha : (spectrum ℝ a).Nonempty
x : ℝ
hx₁ : x ∈ spectrum ℝ a
hx₂ : ↑‖x‖₊ = spectralRadius ℝ a
h : |x| = -x
⊢ -|x| ∈ spectrum ℝ a

Tactic:
simpa [h]

Goal state after: