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

Goal state before:
pp: case refine_3.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
r : NNReal
hr : r * ‖a‖₊⁻¹ < 1
ha : 0 < ‖a‖₊
k : 𝕜
hk₁ : r * ‖a‖₊⁻¹ * ‖a‖₊⁻¹ < ‖k‖₊
hk₂ : ‖k‖₊ < 1 * ‖a‖₊⁻¹
⊢ r * (‖a‖₊ * ‖a‖₊)⁻¹ < ‖k‖₊

Tactic:
mul_inv,

Goal state after:
case refine_3.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
r : NNReal
hr : r * ‖a‖₊⁻¹ < 1
ha : 0 < ‖a‖₊
k : 𝕜
hk₁ : r * ‖a‖₊⁻¹ * ‖a‖₊⁻¹ < ‖k‖₊
hk₂ : ‖k‖₊ < 1 * ‖a‖₊⁻¹
⊢ r * (‖a‖₊⁻¹ * ‖a‖₊⁻¹) < ‖k‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
f : ContinuousMapZero (↑s) 𝕜
hf : f ∈ ↑(adjoin 𝕜 {ContinuousMapZero.id s})
⊢ ContinuousMapZero.id s ∈ ↑(NonUnitalStarAlgHom.equalizer φ ψ)

Tactic:
exact h

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
g : C(X, ℝ)
H : g ∈ Metric.ball f ε
m : g ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε

Tactic:
rw [Metric.mem_ball, dist_eq_norm] at H

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
g : C(X, ℝ)
H : ‖g - f‖ < ε
m : g ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
A : Subalgebra ℝ C(X, ℝ)
hA : A.SeparatesPoints
f : C(X, ℝ)
K : Set X
hK : IsCompact K
ε : ℝ
pos : 0 < ε
restrict_on_K : C(X, ℝ) →⋆ₐ[ℝ] C(↑K, ℝ) := compStarAlgHom' ℝ ℝ { toFun := Subtype.val, continuous_toFun := ⋯ }
AK : Subalgebra ℝ C(↑K, ℝ) := Subalgebra.map (↑restrict_on_K) A
x y : ↑K
hxy : x ≠ y
w✝ : X → ℝ
hg_sep : w✝ ↑x ≠ w✝ ↑y
g : C(X, ℝ)
hg1 : g ∈ ↑A
hg2 : (fun f ↦ ⇑f) g = w✝
⊢ ∃ a ∈ AK, a x ≠ a y

Tactic:
use restrict_on_K g

Goal state after:
case h
X : Type u_2
inst✝ : TopologicalSpace X
A : Subalgebra ℝ C(X, ℝ)
hA : A.SeparatesPoints
f : C(X, ℝ)
K : Set X
hK : IsCompact K
ε : ℝ
pos : 0 < ε
restrict_on_K : C(X, ℝ) →⋆ₐ[ℝ] C(↑K, ℝ) := ⋯
AK : Subalgebra ℝ C(↑K, ℝ) := ⋯
x y : ↑K
hxy : x ≠ y
w✝ : X → ℝ
hg_sep : w✝ ↑x ≠ w✝ ↑y
g : C(X, ℝ)
hg1 : g ∈ ↑A
hg2 : (fun f ↦ ⇑f) g = w✝
⊢ restrict_on_K g ∈ AK ∧ (restrict_on_K g) x ≠ (restrict_on_K g) y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝¹] ⇑φ
⊢ Continuous[_, inst✝¹] ⇑φ.realContinuousMapZeroOfNNReal

Tactic:
by
  dsimp [realContinuousMapZeroOfNNReal]
  fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : CompactSpace ↑s
⊢ (adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}).topologicalClosure = ⊤

Tactic:
← polynomialFunctions.starClosure_topologicalClosure,

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : CompactSpace ↑s
⊢ (adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}).topologicalClosure =
    (polynomialFunctions s).starClosure.topologicalClosure
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakBilin

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
E : Type u_4
F : Type u_5
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : CommSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommMonoid F
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
l : Filter α
f : α → WeakBilin B
x : WeakBilin B
hB : Function.Injective ⇑B
⊢ Tendsto f l (𝓝 x) ↔ ∀ (y : F), Tendsto (fun i ↦ (B (f i)) y) l (𝓝 ((B x) y))

Tactic:
rw [← tendsto_pi_nhds, (isEmbedding hB).tendsto_nhds_iff]
  rfl

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
γ : Type u_5
inst✝ : TopologicalSpace γ
i : γ → β
hf : UpperHemicontinuousWithinAt f s x
hi : IsInducing i
h_cl : IsClosed[inst✝¹] (range i)
⊢ UpperHemicontinuousWithinAt (fun x ↦ i ⁻¹' f x) s x

Tactic:
refine .of_forall_isOpen fun u hu hifu ↦ ?_

Goal state after:
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
γ : Type u_5
inst✝ : TopologicalSpace γ
i : γ → β
hf : UpperHemicontinuousWithinAt f s x
hi : IsInducing i
h_cl : IsClosed[inst✝¹] (range i)
u : Set γ
hu : IsOpen[inst✝] u
hifu : i ⁻¹' f x ⊆ u
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, i ⁻¹' f x' ⊆ u
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ∈ ⋂₀ (powerset '' u) ∩ ⋂₀ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v)
⊢ ⋂₀ (powerset '' u) = 𝒫 U

Tactic:
ext; simp [U]

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
x✝ : CompleteSpace (NonemptyCompacts α)
f : Filter α
hf : Cauchy f
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
U : Set α
⊢ x ∈ U ∧ IsOpen[inst✝.toTopologicalSpace] U → U ∈ f

Tactic:
⟨hxU, hU⟩

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
x✝ : CompleteSpace (NonemptyCompacts α)
f : Filter α
hf : Cauchy f
K : NonemptyCompacts α
hK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K
x : α
hx : x ∈ ↑K
U : Set α
hxU : x ∈ U
hU : IsOpen[inst✝.toTopologicalSpace] U
⊢ U ∈ f
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
F : Set α
h : IsClosed[inst✝] F
⊢ IsClosed {K | (↑K ∩ Fᶜᶜ).Nonempty}

Tactic:
Set.inter_compl_nonempty_iff,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
F : Set α
h : IsClosed[inst✝] F
⊢ IsClosed {K | ¬↑K ⊆ Fᶜ}
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: A : Type u_2
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace ℝ A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : IsScalarTower ℝ A A
inst✝ : CompleteSpace A
⊢ UpperHemicontinuous (quasispectrum ℝ≥0)

Tactic:
obtain ⟨⟨h₁, -⟩, h₂⟩ := NNReal.isClosedEmbedding_coe

Goal state after:
A : Type u_2
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace ℝ A
inst✝² : SMulCommClass ℝ A A
inst✝¹ : IsScalarTower ℝ A A
inst✝ : CompleteSpace A
h₂ : IsClosed (range NNReal.toReal)
h₁ : IsInducing NNReal.toReal
⊢ UpperHemicontinuous (quasispectrum ℝ≥0)
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : SMul 𝕜 E
inst✝² : AddCommMonoid E
inst✝¹ : PartialOrder 𝕜
inst✝ : TopologicalSpace E
s : Set E
⊢ (closedAbsConvexHull 𝕜) (closure[inst✝] s) ⊆ (closedAbsConvexHull 𝕜) s

Tactic:
simpa using ((closedAbsConvexHull 𝕜).monotone
      (closure_subset_closedAbsConvexHull (𝕜 := 𝕜) (E := E)))

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : PartialOrder 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : LocallyConvexSpace 𝕜 E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ZeroLEOneClass 𝕜
⊢ (𝓝 0).HasBasis (fun s ↦ 0 ∈ s ∧ IsOpen[inst✝⁴] s ∧ AbsConvex 𝕜 s) id

Tactic:
by
  refine (nhds_hasBasis_absConvex 𝕜 E).to_hasBasis ?_ ?_
  · intro s ⟨hs_nhds, hs_balanced, hs_convex⟩
    refine ⟨interior s, ?_, interior_subset⟩
    exact
      ⟨mem_interior_iff_mem_nhds.mpr hs_nhds, isOpen_interior,
        hs_balanced.interior (mem_interior_iff_mem_nhds.mpr hs_nhds), hs_convex.interior⟩
  intro s ⟨hs_zero, hs_open, hs_balanced, hs_convex⟩
  exact ⟨s, ⟨hs_open.mem_nhds hs_zero, hs_balanced, hs_convex⟩, rfl.subset⟩

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

Goal state before:
pp: case mpr
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
t𝕜 : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : T0Space 𝕜
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
x✝ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
φ_cont : Continuous[⨅ i, induced (⇑(f i)) t𝕜, t𝕜] ⇑φ
x : E
hx : x ∈ ⨅ i, (f i).ker
⊢ x ∈ φ.ker

Tactic:
Submodule.mem_iInf,

Goal state after:
case mpr
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
t𝕜 : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : T0Space 𝕜
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
x✝ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
φ_cont : Continuous[⨅ i, induced (⇑(f i)) t𝕜, t𝕜] ⇑φ
x : E
hx : ∀ (i : ι), x ∈ (f i).ker
⊢ x ∈ φ.ker
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp : ∀ (x : ↥S), f x ≤ p ↑x
⊢ ?m.34

Tactic:
refine exists_extension_of_le_sublinear ⟨S, f⟩ p (fun _ hc _ => ?_) ?_ hp
    · simp [map_smul_eq_mul, abs_of_nonneg hc.le]
    · exact fun x y => map_add_le_add p x y

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

Goal state before:
pp: X : Type u_1
R : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
f : C(X, R)
h : ∀ (x : X), IsUnit (f x)
⊢ Continuous fun x ↦ ⋯.unit

Tactic:
by
  refine
    continuous_induced_rng.2
      (Continuous.prodMk f.continuous
        (MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))
  have := NormedRing.inverse_continuousAt (h x).unit
  simp only
  simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢
  exact this.comp (f.continuousAt x)

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

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

Tactic:
rw [dist_comm, dist_homothety_self]

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 ≤ 1

Tactic:
induction n generalizing c with
  | zero => exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one
  | succ n ihn =>
    simp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, ← div_eq_inv_mul]
    have := add_le_add (ihn (left c)) (ihn (right c))
    norm_num at this
    exact Iff.mpr (div_le_one zero_lt_two) this

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case zero
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
c : CU P
⊢ approx 0 c x ≤ approx (0 + 1) c x

Tactic:
simp only [approx, right_U, right_le_midpoint]
    exact (approx_mem_Icc_right_left c 0 x).2

Goal state after:
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 (r' * (a - b) + b) ≤ r' • (f a - f b) + f b ↔ (r' * (b - a) * (b - a)⁻¹) • (f b - f a) ≤ f b - f (r' * (a - b) + b)

Tactic:
mul_inv_cancel_right₀,

Goal state after:
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 (r' * (a - b) + b) ≤ r' • (f a - f b) + f b ↔ r' • (f b - f a) ≤ f b - f (r' * (a - b) + b)

case h
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)
⊢ b - a ≠ 0
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
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
⊢ 0 ≤ f x ∧ f x ≤ 1

Tactic:
simpa using h'f x

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

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
inst✝¹ : T2Space X
inst✝ : RCLike 𝕜
⊢ Bijective ⇑(continuousMapEval X 𝕜)

Tactic:
by
  refine ⟨fun x y hxy => ?_, fun φ => ?_⟩
  · contrapose! hxy
    rcases exists_continuous_zero_one_of_isClosed (isClosed_singleton : _root_.IsClosed {x})
        (isClosed_singleton : _root_.IsClosed {y}) (Set.disjoint_singleton.mpr hxy) with
      ⟨f, fx, fy, -⟩
    rw [DFunLike.ne_iff]
    use (⟨fun (x : ℝ) => (x : 𝕜), RCLike.continuous_ofReal⟩ : C(ℝ, 𝕜)).comp f
    simpa only [continuousMapEval_apply_apply, ContinuousMap.comp_apply, coe_mk, Ne,
      RCLike.ofReal_inj] using
      ((fx (Set.mem_singleton x)).symm ▸ (fy (Set.mem_singleton y)).symm ▸ zero_ne_one : f x ≠ f y)
  · obtain ⟨x, hx⟩ := (ideal_isMaximal_iff (RingHom.ker φ)).mp inferInstance
    refine ⟨x, CharacterSpace.ext_ker <| Ideal.ext fun f => ?_⟩
    simpa only [RingHom.mem_ker, continuousMapEval_apply_apply, mem_idealOfSet_compl_singleton,
      RingHom.mem_ker] using SetLike.ext_iff.mp hx f

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

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺ * x⁻) a = cfcₙ 0 a

Tactic:
refine cfcₙ_congr (fun x _ ↦ ?_)

Goal state after:
case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ x⁺ * x⁻ = 0 x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻ * x⁺) a = cfcₙ 0 a

Tactic:
refine cfcₙ_congr (fun x _ ↦ ?_)

Goal state after:
case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
x✝ : x ∈ quasispectrum ℝ a
⊢ x⁻ * x⁺ = 0 x
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:
by
  rw [ContinuousOn] at ha_cont ⊢
  refine fun x hx ↦ (ha_cont x hx).cfc (hs x hx) f hx ?_ ?_ (hf x hx)
  all_goals filter_upwards [ha x hx, self_mem_nhdsWithin] with x hx hxt
  exacts [hx, ha' x hxt]

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
a : A
⊢ ⟪a •> x - ‖x‖ ^ 2 •> y, a •> x - ‖x‖ ^ 2 •> y⟫_A =
    a * ⟪x, x⟫_A * star a - ‖x‖ ^ 2 •> (a * ⟪y, x⟫_A) - ‖x‖ ^ 2 •> (⟪x, y⟫_A * star a) + ‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A

Tactic:
by
                      simp only [inner_sub_right, inner_op_smul_right, inner_sub_left,
                        inner_op_smul_left, inner_smul_left_real, mul_sub, mul_smul_comm,
                        inner_smul_right_real, smul_sub, mul_assoc]
                      abel

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

Goal state before:
pp: case h₂
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪y, x⟫‖ ≤ ‖y‖ * ‖x‖

Tactic:
exact norm_inner_le E

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

Goal state before:
pp: case h₂
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖

Tactic:
exact norm_inner_le E

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
⊢ ‖x‖ ≤ ↑↑(Fintype.card ι) * ‖(linearEquiv ℂ A ((i : ι) → E i)) x‖

Tactic:
by
    simp only [NNReal.coe_natCast, linearEquiv_apply]
    calc ‖x‖ ≤ ∑ i, ‖x i‖ := pi_norm_le_sum_norm x
      _ ≤ ∑ _, ‖⇑x‖ := Finset.sum_le_sum fun _ _ ↦ norm_le_pi_norm ..
      _ ≤ Fintype.card ι * ‖⇑x‖ := by simp

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

Goal state before:
pp: case refine_3
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
this : ∀ (a b : A), 0 ≤ a → 0 ≤ b → ‖a‖ < 1 → ‖b‖ < 1 → a ≤ cfcₙ f (cfcₙ g a + cfcₙ g b)
a : A
ha₁ : 0 ≤ a
b : A
hb₁ : 0 ≤ b
ha₂ : ‖a‖ < 1
hb₂ : ‖b‖ < 1
⊢ (fun x1 x2 ↦ x1 ≤ x2) b (cfcₙ f (cfcₙ g a + cfcₙ g b))

Tactic:
exact add_comm (cfcₙ g a) (cfcₙ g b) ▸ this b a hb₁ ha₁ hb₂ ha₂

Goal state after:
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:
linfty_opNNNorm_def,

Goal state after:
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
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖replicateCol ι v i j‖₊) = ‖v‖₊
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
⊢ ‖diagonal v‖₊ = ‖v‖₊

Tactic:
linfty_opNNNorm_def,

Goal state after:
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‖₊) = ‖v‖₊
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:
linfty_opNNNorm_def,

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✝ : NonUnitalSeminormedRing α
A : Matrix l m α
B : Matrix m n α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖(A * B) i j‖₊) ≤
    (Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊) * Finset.univ.sup fun i ↦ ∑ j, ‖B i j‖₊
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
⊢ Summable fun n ↦ 2 ^ (-↑n) • x n

Tactic:
refine Summable.of_norm ?_

Goal state after:
A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
⊢ Summable fun a ↦ ‖2 ^ (-↑a) • x a‖
Module: Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap

Goal state before:
pp: case mk.mk
A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : PartialOrder A₂
inst✝¹ : StarOrderedRing A₁
inst✝ : StarOrderedRing A₂
toLinearMap✝¹ : A₁ →ₗ[ℂ] A₂
map_cstarMatrix_nonneg'✝¹ : ∀ (k : ℕ) (M : CStarMatrix (Fin k) (Fin k) A₁), 0 ≤ M → 0 ≤ M.map ⇑toLinearMap✝¹
toLinearMap✝ : A₁ →ₗ[ℂ] A₂
map_cstarMatrix_nonneg'✝ : ∀ (k : ℕ) (M : CStarMatrix (Fin k) (Fin k) A₁), 0 ≤ M → 0 ≤ M.map ⇑toLinearMap✝
h :
  (fun f ↦ f.toFun) { toLinearMap := toLinearMap✝¹, map_cstarMatrix_nonneg' := map_cstarMatrix_nonneg'✝¹ } =
    (fun f ↦ f.toFun) { toLinearMap := toLinearMap✝, map_cstarMatrix_nonneg' := map_cstarMatrix_nonneg'✝ }
⊢ ⇑toLinearMap✝¹ = ⇑toLinearMap✝

Tactic:
exact h

Goal state after: