mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ν ≤ n
⊢ (-1) ^ (n - ν) * eval (↑(n - (n - ν - 1))) (ascPochhammer R (n - ν)) =
    (-1) ^ (n - ν) * eval (↑ν + 1) (ascPochhammer R (n - ν))

Tactic:
obtain rfl | h' := h.eq_or_lt

Goal state after:
case inl
R : Type u_1
inst✝ : CommRing R
ν : ℕ
h : ν ≤ ν
⊢ (-1) ^ (ν - ν) * eval (↑(ν - (ν - ν - 1))) (ascPochhammer R (ν - ν)) =
    (-1) ^ (ν - ν) * eval (↑ν + 1) (ascPochhammer R (ν - ν))

case inr
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
h : ν ≤ n
h' : ν < n
⊢ (-1) ^ (n - ν) * eval (↑(n - (n - ν - 1))) (ascPochhammer R (n - ν)) =
    (-1) ^ (n - ν) * eval (↑ν + 1) (ascPochhammer R (n - ν))
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
c : ContinuousOn f (Set.Icc a b)
ε : ℝ
pos : 0 < ε
⊢ ∃ p, ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε

Tactic:
by
  let f' : C(Set.Icc a b, ℝ) := ⟨fun x => f x, continuousOn_iff_continuous_domRestrict.mp c⟩
  obtain ⟨p, b⟩ := exists_polynomial_near_continuousMap a b f' ε pos
  use p
  rw [norm_lt_iff _ pos] at b
  intro x m
  exact b ⟨x, m⟩

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

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := fun x y ↦ {z | f z - ε < (g x y) z}
x y : X
⊢ y ∈ U x y

Tactic:
rw [Set.mem_ofPred_eq, w₂]
      exact sub_lt_self _ pos

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

Goal state before:
pp: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal =
    φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal =
    φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
      (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)

Tactic:
← sub_eq_zero

Goal state after:
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal) =
    0
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal -
      (φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
        (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)) =
    0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
this : φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal = φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal = φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)

Tactic:
← sub_eq_zero

Goal state after:
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f g : C(X, ℝ)
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) = 0
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
f g : C(X, ℝ)₀
this : φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal = φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal = φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)

Tactic:
← sub_eq_zero

Goal state after:
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
f g : C(X, ℝ)₀
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) = 0
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
f g : C(X, ℝ)₀
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal =
    φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal =
    φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
      (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)

Tactic:
← sub_eq_zero

Goal state after:
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
f g : C(X, ℝ)₀
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal) =
    0
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal -
      (φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
        (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)) =
    0
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : (algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : ¬(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖

Tactic:
· simp only [h, norm_zero]
        exact norm_nonneg _

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CStarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : StarModule 𝕜 E
inst✝ : CStarRing 𝕜
x✝ x : Unitization 𝕜 E
h : ¬(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2 = 0
⊢ ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2‖ ≤
    ‖(algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1⋆ + (mul 𝕜 E) x.toProd.2⋆‖
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) ↔ Tendsto (fun i ↦ ⇑(B (f i))) l (𝓝 ⇑(B x))

Tactic:
(isEmbedding hB).tendsto_nhds_iff

Goal state after:
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 ((fun x y ↦ (B x) y) ∘ f) l (𝓝 fun y ↦ (B x) y) ↔ Tendsto (fun i ↦ ⇑(B (f i))) l (𝓝 ⇑(B x))
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: A : Type u_1
inst✝⁴ : Semiring A
inst✝³ : Algebra ℝ A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set ℝ
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, ℝ) →ₐ[ℝ] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
⊢ ⊤ ≤ φ.equalizer ψ

Tactic:
← polynomialFunctions.topologicalClosure s

Goal state after:
A : Type u_1
inst✝⁴ : Semiring A
inst✝³ : Algebra ℝ A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set ℝ
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, ℝ) →ₐ[ℝ] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
⊢ (polynomialFunctions s).topologicalClosure ≤ φ.equalizer ψ
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✝²⁹ : CommSemiring R
inst✝²⁸ : Nontrivial R
inst✝²⁷ : StarRing R
inst✝²⁶ : MetricSpace R
inst✝²⁵ : IsTopologicalSemiring R
inst✝²⁴ : ContinuousStar R
inst✝²³ : CommRing S
inst✝²² : Algebra R S
inst✝²¹ : NonUnitalRing A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : Module R A
inst✝¹⁷ : IsScalarTower R A A
inst✝¹⁶ : SMulCommClass R A A
inst✝¹⁵ : NonUnitalRing B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Module R B
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : SMulCommClass R B B
inst✝⁹ : Module S A
inst✝⁸ : Module S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : NonUnitalContinuousFunctionalCalculus R A p
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMapZero.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (quasispectrum R a)
hf₀ : f 0 = 0
hφ : Continuous[inst✝¹⁹, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₙₐ[R] B := NonUnitalStarAlgHom.restrictScalars R ↑φ
this : Continuous[inst✝¹⁹, inst✝¹³] ⇑ψ
h_spec : quasispectrum R (φ a) ⊆ quasispectrum R a
hψa : q (ψ a)
⊢ φ (cfcₙ f a) = cfcₙ f (φ a)

Tactic:
let ι : C(quasispectrum R (ψ a), quasispectrum R a)₀ :=
    ⟨⟨Set.inclusion h_spec, continuous_id.subtype_map h_spec⟩, rfl⟩

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✝²⁹ : CommSemiring R
inst✝²⁸ : Nontrivial R
inst✝²⁷ : StarRing R
inst✝²⁶ : MetricSpace R
inst✝²⁵ : IsTopologicalSemiring R
inst✝²⁴ : ContinuousStar R
inst✝²³ : CommRing S
inst✝²² : Algebra R S
inst✝²¹ : NonUnitalRing A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : Module R A
inst✝¹⁷ : IsScalarTower R A A
inst✝¹⁶ : SMulCommClass R A A
inst✝¹⁵ : NonUnitalRing B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Module R B
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : SMulCommClass R B B
inst✝⁹ : Module S A
inst✝⁸ : Module S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : NonUnitalContinuousFunctionalCalculus R A p
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMapZero.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (quasispectrum R a)
hf₀ : f 0 = 0
hφ : Continuous[inst✝¹⁹, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₙₐ[R] B := NonUnitalStarAlgHom.restrictScalars R ↑φ
this : Continuous[inst✝¹⁹, inst✝¹³] ⇑ψ
h_spec : quasispectrum R (φ a) ⊆ quasispectrum R a
hψa : q (ψ a)
ι : C(↑(quasispectrum R (ψ a)), ↑(quasispectrum R a))₀ :=
  { toFun := Set.inclusion h_spec, continuous_toFun := ⋯, map_zero' := ⋯ }
⊢ φ (cfcₙ f a) = cfcₙ f (φ a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖a‖₊ ^ n.succ * r ^ (n + 1) ≤ 1

Tactic:
by_cases h : ‖a‖₊ = 0

Goal state after:
case pos
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
h : ‖a‖₊ = 0
⊢ ‖a‖₊ ^ n.succ * r ^ (n + 1) ≤ 1

case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
h : ¬‖a‖₊ = 0
⊢ ‖a‖₊ ^ n.succ * r ^ (n + 1) ≤ 1
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
⊢ ‖y • a‖ < 1

Tactic:
by_cases h : ‖a‖₊ = 0

Goal state after:
case pos
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ‖a‖₊ = 0
⊢ ‖y • a‖ < 1

case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y • a‖ < 1
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
hf : UniformContinuous f
U : Set (β × β)
hU : U ∈ 𝓤 β
s t : Set α
h₁ : (s, t).1 ⊆ SetRel.preimage ((fun x ↦ (f x.1, f x.2)) ⁻¹' U) (s, t).2
h₂ : (s, t).2 ⊆ SetRel.image ((fun x ↦ (f x.1, f x.2)) ⁻¹' U) (s, t).1
⊢ s ⊆ f ⁻¹' SetRel.preimage U (f '' t) ∧ t ⊆ f ⁻¹' SetRel.image U (f '' s)

Tactic:
exact ⟨h₁.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩,
    h₂.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩⟩

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 ≤ c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖cfc f a‖ ≤ c

Tactic:
obtain (_ | _) := subsingleton_or_nontrivial A
  · simpa [Subsingleton.elim (cfc f a) 0]
  · refine cfc_cases (‖·‖ ≤ c) a f (by simpa) fun hf ha ↦ ?_
    simp only [← cfc_apply f a, isLUB_le_iff (IsGreatest.norm_cfc f a hf ha |>.isLUB)]
    rintro - ⟨x, hx, rfl⟩
    exact h x hx

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : E
hu : u ∈ unitary E
z : 𝕜
hz : z ∈ σ 𝕜 u
⊢ ‖z‖ = 1

Tactic:
simpa using spectrum.subset_circle_of_unitary hu hz

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
⊢ Continuous fun p ↦ p.1 ×ˢ p.2

Tactic:
rw [continuous_induced_rng, continuous_generateFrom_iff]
  rintro _ (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩)
  · rw [isOpen_iff_forall_mem_open]
    intro ⟨K, L⟩ (hKL : (K : Set α) ×ˢ (L : Set β) ⊆ U)
    obtain ⟨V, W, hV, hW, hK, hL, hVW⟩ := generalized_tube_lemma K.isCompact L.isCompact hU hKL
    exact ⟨_, fun ⟨K', L'⟩ ⟨hK', hL'⟩ => (Set.prod_mono hK' hL').trans hVW,
      (isOpen_subsets_of_isOpen hV).prod (isOpen_subsets_of_isOpen hW), hK, hL⟩
  · rw [isOpen_iff_forall_mem_open]
    intro ⟨K, L⟩ ⟨⟨x, y⟩, ⟨(hx : x ∈ K), (hy : y ∈ L)⟩, hxy⟩
    obtain ⟨V, W, hV, hW, hxV, hyW, hVW⟩ := isOpen_prod_iff.mp hU x y hxy
    grw [preimage_ofPred_eq, ← hVW]
    simp_rw [Function.comp_apply, coe_prod, prod_inter_prod, prod_nonempty_iff]
    exact ⟨_, .rfl,
      (isOpen_inter_nonempty_of_isOpen hV).prod (isOpen_inter_nonempty_of_isOpen hW),
      ⟨x, hx, hxV⟩, ⟨y, hy, hyW⟩⟩

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

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

Tactic:
rw [forall_mem_image] at hsu

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
U : Set (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
⊢ IsPreconnected U

Tactic:
suffices IsPreconnected (U ∩ {K | (K : Set α).Finite}) by
    refine this.subset_closure inter_subset_left ?_
    grw [← (basis.isOpen hU).inter_closure, dense_setOfPred_finite.closure_eq, inter_univ]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
U : Set (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
⊢ IsPreconnected (U ∩ {K | (↑K).Finite})
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : α) in cocompact α, dist (f x) 0 < ε

Tactic:
obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f <| half_pos hε)

Goal state after:
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
ε : ℝ
hε : ε > 0
g : α →C₀ β
hg : g.toBCF ∈ ball f (ε / 2)
⊢ ∀ᶠ (x : α) in cocompact α, dist (f x) 0 < ε
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
r : ℝ
hr : ‖r‖ ≤ 1
y : E
hy : y ∈ s
ha : (fun x ↦ r • x) y ∈ balancedHull ℝ s
this : 0 ≤ 1 + r
⊢ (fun x ↦ r • x) y ∈ segment ℝ y (-y)

Tactic:
have : 0 ≤ 1 - r := sub_nonneg.2 (le_of_abs_le hr)

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
r : ℝ
hr : ‖r‖ ≤ 1
y : E
hy : y ∈ s
ha : (fun x ↦ r • x) y ∈ balancedHull ℝ s
this✝ : 0 ≤ 1 + r
this : 0 ≤ 1 - r
⊢ (fun x ↦ r • x) y ∈ segment ℝ y (-y)
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := ⋯
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
h_deriv₂ :
  ∀ (a b : A),
    deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2
h₄ : a ^ 2 + 2 • (-a * star a) + star a ^ 2 = star a ^ 2 + 2 • (star a * -a) + a ^ 2
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-(x • a)) * NormedSpace.exp (x • star a) =
      NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))
h₃ :
  deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))) y) 0 =
    a ^ 2 + 2 • (-a * star a) + star a ^ 2
⊢ (fun a ↦ 2 • a) (star a * a) = (fun a ↦ 2 • a) (a * star a)

Tactic:
← sub_eq_zero

Goal state after:
case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
h_deriv₂ :
  ∀ (a b : A),
    deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2
h₄ : a ^ 2 + 2 • (-a * star a) + star a ^ 2 - (star a ^ 2 + 2 • (star a * -a) + a ^ 2) = 0
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-(x • a)) * NormedSpace.exp (x • star a) =
      NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))
h₃ :
  deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))) y) 0 =
    a ^ 2 + 2 • (-a * star a) + star a ^ 2
⊢ (fun a ↦ 2 • a) (star a * a) - (fun a ↦ 2 • a) (a * star a) = 0
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: case neg
𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
hx : ¬‖x‖ = 0
⊢ ∃ g, ‖g‖ ≤ 1 ∧ g x = ↑‖x‖

Tactic:
· obtain ⟨g, hg⟩ := exists_dual_vector 𝕜 x hx
    exact ⟨g, hg.left.le, hg.right⟩

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

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_4
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
| closedBall 0 r

Tactic:
← inv_inv r

Goal state after:
𝕜 : Type u_3
E : Type u_4
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
| closedBall 0 r⁻¹⁻¹