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✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ g x
hf : ContinuousOn f (σₙ R a)
hg : ContinuousOn g (σₙ R a)
hf0 : f 0 = 0
hg0 : g 0 = 0
ha : p a
⊢ cfcₙ f a ≤ cfcₙ g a

Tactic:
rw [cfcₙ_apply f a, cfcₙ_apply g a]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ g x
hf : ContinuousOn f (σₙ R a)
hg : ContinuousOn g (σₙ R a)
hf0 : f 0 = 0
hg0 : g 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } ≤
    (cfcₙHom ha) { toFun := (σₙ R a).domRestrict g, continuous_toFun := ⋯, map_zero' := ⋯ }
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
⊢ f y - ε < (g x y) y

Tactic:
w₂

Goal state after:
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
⊢ f y - ε < f y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

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

Tactic:
sub_zero

Goal state after:
X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ f = φ f
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case 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 x : E
hx : x ∈ Metric.closedBall 0 1
⊢ (fun x ↦ ‖((mul 𝕜 E) a) x‖₊) x ≤ ‖a‖₊

Tactic:
exact
        ((mul 𝕜 E a).unit_le_opNorm x <| mem_closedBall_zero_iff.mp hx).trans
          (opNorm_mul_apply_le 𝕜 E a)

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
f g : C(X, ℝ)₀
⊢ (f + g).toNNReal + (-f).toNNReal + (-g).toNNReal = (-(f + g)).toNNReal + f.toNNReal + g.toNNReal

Tactic:
by
  apply toContinuousMap_injective
  simpa only [map_add, map_mul, map_neg, toContinuousMapHom_toNNReal]
    using! (f : C(X, ℝ)).toNNReal_add_add_neg_add_neg_eq g

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

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ f - 0 = φ f

Tactic:
sub_zero

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ f = φ f
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case h₂
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
y : E
hy : y ∈ Metric.closedBall 0 1
⊢ ‖↑((mul 𝕜 E) (x⋆ * x).toProd.2) y‖ ≤ ‖(x⋆ * x).toProd.2‖

Tactic:
exact (unit_le_opNorm _ y <| mem_closedBall_zero_iff.1 hy).trans (opNorm_mul_apply_le _ _ _)

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

Goal state before:
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
this : ‖((splitMul 𝕜 E) x).2‖ ^ 2 ≤ ‖((splitMul 𝕜 E) x⋆).2‖ * ‖((splitMul 𝕜 E) x).2‖
⊢ ‖(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:
sq

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
this : ‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖ * ‖((splitMul 𝕜 E) x).2‖
⊢ ‖(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.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case mem
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝¹ f✝ f : C(↑s, 𝕜)
hf : f = (toContinuousMapOnAlgHom s) X ∨ star f = (toContinuousMapOnAlgHom s) X
⊢ p f

Tactic:
star_eq_iff_star_eq,

Goal state after:
case mem
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝¹ f✝ f : C(↑s, 𝕜)
hf : f = (toContinuousMapOnAlgHom s) X ∨ star ((toContinuousMapOnAlgHom s) X) = f
⊢ p f
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 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 : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
⊢ ‖((splitMul 𝕜 E) x⋆).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x).2‖ ^ 2

Tactic:
sq

Goal state after:
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 : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
⊢ ‖((splitMul 𝕜 E) x⋆).2‖ * ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x).2‖ * ‖((splitMul 𝕜 E) x).2‖
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 = ψ

Tactic:
suffices (⊤ : Subalgebra ℝ C(s, ℝ)) ≤ AlgHom.equalizer φ ψ from
    AlgHom.ext fun x => this (by trivial)
  rw [← polynomialFunctions.topologicalClosure s]
  exact Subalgebra.topologicalClosure_minimal
    (polynomialFunctions.le_equalizer s φ ψ h) (isClosed_eq hφ hψ)

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

Goal state before:
pp: case mul
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : φ (ContinuousMapZero.id s) * a = 0
hmul_star_id : φ (star (ContinuousMapZero.id s)) * a = 0
hφ : Continuous ⇑φ
f✝ g✝ : C(↑s, 𝕜)₀
a✝ : φ f✝ * a = 0
h : φ g✝ * a = 0
⊢ φ (f✝ * g✝) * a = 0

Tactic:
| mul _ _ _ h =>

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
s : Set α
inst✝ : TopologicalSpace β
⊢ UpperHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' Iic u ∈ 𝓝[s] x

Tactic:
simp [upperHemicontinuousOn_iff, upperHemicontinuousWithinAt_iff_preimage_Iic]

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
x₀ : α
inst✝ : (𝓝 x₀).IsCountablyGenerated
h : ∀ (x : ℕ → α), Tendsto x atTop (𝓝 x₀) → ∀ y₀ ∈ f x₀, ∃ y, (∀ (n : ℕ), y n ∈ f (x n)) ∧ Tendsto y atTop (𝓝 y₀)
U : Set β
hU : IsOpen[inst✝¹] U
y₀ : β
hy₀f : y₀ ∈ f x₀
hy₀U : y₀ ∈ U
hc : ∃ᶠ (x : α) in 𝓝 x₀, ¬(f x ∩ U).Nonempty
x : ℕ → α
hx : Tendsto x atTop (𝓝 x₀)
hxU : ∀ (n : ℕ), ¬(f (x n) ∩ U).Nonempty
y : ℕ → β
hy_mem : ∀ (n : ℕ), y n ∈ f (x n)
hy_lim : Tendsto y atTop (𝓝 y₀)
n : ℕ
hn : U (y n)
⊢ False

Tactic:
exact hxU n ⟨y n, hy_mem n, hn⟩

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

Goal state before:
pp: case a
α : Type u_1
U V : SetRel α α
⊢ hausdorffEntourage U ○ hausdorffEntourage V ⊆ hausdorffEntourage (U ○ V)

Tactic:
intro ⟨s₁, s₃⟩

Goal state after:
case a
α : Type u_1
U V : SetRel α α
s₁ s₃ : Set α
⊢ (s₁, s₃) ∈ hausdorffEntourage U ○ hausdorffEntourage V → (s₁, s₃) ∈ hausdorffEntourage (U ○ V)
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✝ : CompleteSpace A
a : A
h : (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1
⊢ resolvent a =ᶠ[cobounded 𝕜] fun z ↦ z⁻¹ • resolvent (z⁻¹ • a) 1

Tactic:
filter_upwards [isBounded_singleton (x := 0)] with z hz
      lift z to 𝕜ˣ using Ne.isUnit hz
      simpa [Units.smul_def] using congr(z⁻¹ • $(units_smul_resolvent_self (r := z) (a := a)))

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : ∀ i ∈ 𝓤 α, ∃ x ∈ Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i, x ∈ {s | TotallyBounded s}
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
t : Set α
ht : t ∈ {s | TotallyBounded s}
hst : s ⊆ V.preimage t
u : Set α
hu : u.Finite
htu : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ V}
⊢ V.preimage t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}

Tactic:
htu,

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : ∀ i ∈ 𝓤 α, ∃ x ∈ Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i, x ∈ {s | TotallyBounded s}
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
t : Set α
ht : t ∈ {s | TotallyBounded s}
hst : s ⊆ V.preimage t
u : Set α
hu : u.Finite
htu : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ V}
⊢ V.preimage (⋃ y ∈ u, {x | (x, y) ∈ V}) ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
s t : Set α
inst✝ : T1Space α
⊢ s ⤳ t ↔ s ⊆ t ∧ t ⊆ closure[inst✝¹] s

Tactic:
simp_rw [specializes_iff, specializes_iff_eq, existsAndEq, and_true, ← subset_def]

Goal state after:
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
μ : 𝕜
⊢ μ ∈ frontier (σ 𝕜 x) → μ ∈ σ 𝕜 ↑x

Tactic:
hμ

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
μ : 𝕜
hμ : μ ∈ frontier (σ 𝕜 x)
⊢ μ ∈ σ 𝕜 ↑x
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
⊢ ↑r ≤ liminf (fun i ↦ 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop

Tactic:
convert! h

Goal state after:
A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
⊢ liminf (fun i ↦ 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop = p.radius
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ IsTopologicalBasis ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B})

Tactic:
refine hB.vietoris.isInducing isEmbedding_coe.isInducing
    |>.isTopologicalBasis_of_exists_subset ?_ ?_ <;> simp_rw [forall_mem_image]
  · intro u ⟨hu, huB⟩
    simp_rw [ofPred_and, ofPred_forall]
    exact .inter
      (isOpen_subsets_of_isOpen <| isOpen_sUnion fun U hU => hB.isOpen <| huB hU)
      (hu.isOpen_biInter fun U hU => isOpen_inter_nonempty_of_isOpen <| hB.isOpen <| huB hU)
  · intro ⟨V, u⟩ ⟨hV, hu, huB, huV⟩ K ⟨hKV, hKu⟩
    dsimp only at *
    obtain ⟨w, hwB, hwV⟩ := hB.open_eq_sUnion hV
    rw [hwV] at hKV
    replace hwV := hwV.symm.subset
    wlog hwK : ∀ W ∈ w, ((K : Set α) ∩ W).Nonempty generalizing w
    · refine this {W ∈ w | (↑K ∩ W).Nonempty} (fun x hx => ?_) (by grind) (by grind) (by grind)
      obtain ⟨W, hWw, hxW⟩ := mem_sUnion.mp (hKV hx)
      exact mem_sUnion_of_mem hxW ⟨hWw, x, hx, hxW⟩
    wlog hw : w.Finite generalizing w
    · rw [sUnion_eq_biUnion] at hKV
      obtain ⟨w', _⟩ := K.isCompact.elim_finite_subcover_image (fun W hW => hB.isOpen (hwB hW)) hKV
      apply this w' <;> grind [sUnion_eq_biUnion]
    wlog huw : u ⊆ w generalizing w
    · apply this (u ∪ w) <;> grind [sUnion_union, Finite.union]
    rw [exists_mem_image]
    exists w
    #adaptation_note /-- Before leanprover/lean4#13166, this was just `grind`.
    The new canonicalizer using `isDefEq` less,
    and so does not unify as many of the conditions that `grind` wants to case split on.
    Alternatively `simp; grind` also works here. -/
    grind (splits := 12)

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

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C₀ β
x✝ : α
⊢ (r • f x✝) • g x✝ = r • f x✝ • g x✝

Tactic:
smul_assoc

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C₀ β
x✝ : α
⊢ r • f x✝ • g x✝ = r • f x✝ • g x✝
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : AddMonoid β
inst✝¹ : StarAddMonoid β
inst✝ : ContinuousStar β
f : α →C₀ β
⊢ Tendsto (fun x ↦ star (f x)) (cocompact α) (𝓝 0)

Tactic:
by
        simpa only [star_zero] using! (continuous_star.tendsto (0 : β)).comp (zero_at_infty f)

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
hv_open : ∀ (n : ℕ), IsOpen[inst✝⁵] (v n)
hv_nhds : ∀ (n : ℕ), v n ∈ 𝓝 0
hv_basis : (𝓝 0).HasAntitoneBasis v
m : ℕ
⊢ ∀ᶠ (n : ℕ) in Filter.atTop, v n + v n ⊆ v m

Tactic:
obtain ⟨W, hW_open, hW_zero, hW_add⟩ := exists_open_nhds_zero_add_subset (hv_nhds m)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
hv_open : ∀ (n : ℕ), IsOpen[inst✝⁵] (v n)
hv_nhds : ∀ (n : ℕ), v n ∈ 𝓝 0
hv_basis : (𝓝 0).HasAntitoneBasis v
m : ℕ
W : Set E
hW_open : IsOpen[inst✝⁵] W
hW_zero : 0 ∈ W
hW_add : W + W ⊆ v m
⊢ ∀ᶠ (n : ℕ) in Filter.atTop, v n + v n ⊆ v m
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set (Set α)⦄,
    x ∈ {u | u.Finite ∧ u ⊆ B} →
      ∀ x_1 ∈ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty},
        ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
          x_1 ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty}

Tactic:
intro u ⟨hu, huB⟩ K ⟨hK₁, hK₂⟩
    rcases eq_empty_or_nonempty u with rfl | hu'
    · absurd K.nonempty.mono hK₁; simp
    rw [exists_mem_image]
    exists u

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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : NormedField 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : Module 𝕜 E
S : Submodule 𝕜 E
f : Dual 𝕜 ↥S
p : Seminorm 𝕜 E
hp : ∀ (x : ↥S), ‖f x‖ ≤ p ↑x
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ : Module ℝ E := restrictScalars ℝ 𝕜 E
this : IsScalarTower ℝ 𝕜 E := IsScalarTower.restrictScalars ℝ 𝕜 E
fr : Dual ℝ ↥S := reLm ∘ₗ ↑ℝ f
g : Dual ℝ E
hg : ∀ (x : ↥S), g ↑x = fr x
hgp : ∀ (x : E), |g x| ≤ (Seminorm.restrictScalars ℝ p) x
x : ↥S
⊢ ↑(g ↑x) - I * ↑(g (I • ↑x)) = f x

Tactic:
← Submodule.coe_smul,

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : NormedField 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : Module 𝕜 E
S : Submodule 𝕜 E
f : Dual 𝕜 ↥S
p : Seminorm 𝕜 E
hp : ∀ (x : ↥S), ‖f x‖ ≤ p ↑x
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ : Module ℝ E := restrictScalars ℝ 𝕜 E
this : IsScalarTower ℝ 𝕜 E := IsScalarTower.restrictScalars ℝ 𝕜 E
fr : Dual ℝ ↥S := reLm ∘ₗ ↑ℝ f
g : Dual ℝ E
hg : ∀ (x : ↥S), g ↑x = fr x
hgp : ∀ (x : E), |g x| ≤ (Seminorm.restrictScalars ℝ p) x
x : ↥S
⊢ ↑(g ↑x) - I * ↑(g ↑(I • x)) = f x
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: 𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : Nontrivial E
x : E
⊢ ∃ g, ‖g‖ = 1 ∧ g x = ↑‖x‖

Tactic:
by_cases hx : x = 0
  · obtain ⟨y, hy⟩ := exists_norm_ne_zero E
    obtain ⟨g, hg⟩ := exists_dual_vector 𝕜 y hy
    exact ⟨g, hg.left, by simp [hx]⟩
  · exact exists_dual_vector 𝕜 x (norm_ne_zero_iff.mpr hx)

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : StrongDual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : c = 0
⊢ c⁻¹ • x' ∈ polar 𝕜 s

Tactic:
simp only [c_zero, inv_zero, zero_smul]

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : StrongDual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : c = 0
⊢ 0 ∈ polar 𝕜 s
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
| Continuous[t₁, _] ⇑((normSeminorm 𝕜 𝕜).comp φ)

Tactic:
rw [Seminorm.continuous_iff one_pos, nhds_iInf]

Goal state after:
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
| ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, 𝓝 0
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

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

Tactic:
dist_comm

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