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

Goal state before:
pp: A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra ℝ A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : Nontrivial A
a : A
ha : 0 ≤ a
⊢ IsGreatest (σ ℝ≥0 a) ‖a‖₊

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra ℝ A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : Nontrivial A
a : A
ha : 0 ≤ a
⊢ IsGreatest (σ ℝ≥0 a) ‖a‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
ha : 0 ≤ a
⊢ x ≤ ‖a‖₊

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
ha : 0 ≤ a
⊢ x ≤ ‖a‖₊
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z ↦ resolvent a z
⊢ Tendsto (fun z ↦ resolvent a z) (cocompact ℂ) (𝓝 ?m.100)

Tactic:
simpa using

Goal state after:
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z ↦ resolvent a z
⊢ Tendsto (fun z ↦ resolvent a z) (cocompact ℂ) (𝓝 ?m.100)
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
n : ℕ
⊢ spectrum ℂ (a ^ n) = (fun x ↦ x ^ n) '' spectrum ℂ a

Tactic:
simpa using

Goal state after:
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
n : ℕ
⊢ spectrum ℂ (a ^ n) = (fun x ↦ x ^ n) '' spectrum ℂ a
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

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

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
z : ℂ
hz : z ∈ σ ℂ a
⊢ z ∈ σ ℂ a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_1
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
⊢ ‖0‖₊ < c

Tactic:
simpa using

Goal state after:
case refine_1
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x < c
⊢ 0 < c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
⊢ IsGreatest (σₙ ℝ≥0 a) ‖a‖₊

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
⊢ IsGreatest (σₙ ℝ≥0 a) ‖a‖₊
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
x✝ : ℂ
⊢ x✝ ∈ {z | z.im < 0} → x✝ ∈ {z | z.im ≤ 0}

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
x✝ : ℂ
⊢ x✝.im < 0 → x✝.im ≤ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
ha : 0 ≤ a
⊢ x ≤ ‖a‖₊

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
ha : 0 ≤ a
⊢ x ≤ ‖a‖₊
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
x✝ : ℂ
⊢ x✝ ∈ {z | 0 < z.im} → x✝ ∈ {z | 0 ≤ z.im}

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
x✝ : ℂ
⊢ 0 < x✝.im → 0 ≤ x✝.im
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
a : A
ha : p a
this : CompactSpace ↑(σₙ 𝕜 a)
f : ContinuousMapZero (↑(σₙ 𝕜 a)) 𝕜
ι : C(↑(σ 𝕜 a), ↑(σₙ 𝕜 a)) := { toFun := Set.inclusion ⋯, continuous_toFun := ⋯ }
x : 𝕜
hx : x ∈ σₙ 𝕜 a
⊢ x = 0 ∨ x ∈ σ 𝕜 a

Tactic:
simpa using

Goal state after:
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
a : A
ha : p a
this : CompactSpace ↑(σₙ 𝕜 a)
f : ContinuousMapZero (↑(σₙ 𝕜 a)) 𝕜
ι : C(↑(σ 𝕜 a), ↑(σₙ 𝕜 a)) := { toFun := Set.inclusion ⋯, continuous_toFun := ⋯ }
x : 𝕜
hx : x ∈ σₙ 𝕜 a
⊢ x = 0 ∨ x ∈ σ 𝕜 a
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
a : A
h : ∀ (ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B) (x : Unitization ℂ A), ‖ψ x‖₊ ≤ ‖x‖₊
⊢ ‖φ a‖₊ ≤ ‖a‖₊

Tactic:
simpa using

Goal state after:
F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
a : A
h : ∀ (ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B) (x : Unitization ℂ A), ‖ψ x‖₊ ≤ ‖x‖₊
⊢ ‖φ a‖₊ ≤ ‖a‖₊
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✝ : CompactSpace α
⊢ IsCompact univ

Tactic:
simpa using

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✝ : CompactSpace α
⊢ IsCompact univ
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
K L : Compacts α
hK : K ≠ ⊥
hL : IsPreconnected ↑L
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] {K L : Compacts α},
    K ≠ ⊥ → IsPreconnected ↑L → K ≤ L → IsPreconnected (Icc K L)
hKL : ¬K ≤ L
⊢ IsPreconnected (Icc K L)

Tactic:
simpa using

Goal state after:
case inr
α : Type u_1
inst✝ : TopologicalSpace α
K L : Compacts α
hK : K ≠ ⊥
hL : IsPreconnected ↑L
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] {K L : Compacts α},
    K ≠ ⊥ → IsPreconnected ↑L → K ≤ L → IsPreconnected (Icc K L)
hKL : ¬K ≤ L
⊢ IsPreconnected ∅
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x : E
hx : x ∈ s
y : F
hy : y ∈ B.polar s
⊢ ‖(B x) y‖ ≤ 1

Tactic:
exact hy x hx

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x : F
hx : ∀ (i : Set E), i.Finite → i ⊆ s → x ∈ B.polar i
a : E
ha : a ∈ s
⊢ ‖(B a) x‖ ≤ 1

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x : F
hx : ∀ (i : Set E), i.Finite → i ⊆ s → x ∈ B.polar i
a : E
ha : a ∈ s
⊢ ‖(B a) x‖ ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
hf : Differentiable ℂ (expMulMulExp a b x)
this : IsBounded (Set.range (expMulMulExp a b x))
⊢ expMulMulExp a b x z = x

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
hf : Differentiable ℂ (expMulMulExp a b x)
this : IsBounded (Set.range (expMulMulExp a b x))
⊢ NormedSpace.exp (z • star b) * x * NormedSpace.exp (-(z • star a)) = x
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a✝ b x : A
inst✝¹ : IsStarNormal a✝
inst✝ : IsStarNormal b
h : SemiconjBy x a✝ b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a✝) = NormedSpace.exp (z • star b) * x
a : A
⊢ HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a✝ b x : A
inst✝¹ : IsStarNormal a✝
inst✝ : IsStarNormal b
h : SemiconjBy x a✝ b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a✝) = NormedSpace.exp (z • star b) * x
a : A
⊢ HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
this : ∀ (a : A), HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0
⊢ HasDerivAt (fun y ↦ x * NormedSpace.exp (y • star a)) (star b * x) 0

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
key : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
this : ∀ (a : A), HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0
⊢ HasDerivAt (fun y ↦ NormedSpace.exp (y • star b) * x) (star b * x) 0
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
g : α →C₀ β
hg : g.toBCF ∈ ball f (ε / 2)
x : α
hx : dist (g x) 0 < ε / 2
⊢ dist (f x) 0 < ε

Tactic:
calc
      dist (f x) 0 ≤ dist (g.toBCF x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
      _ < dist g.toBCF f + ε / 2 := add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx
      _ ≤ ε := by grw [mem_ball.1 hg, add_halves ε]

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
x✝¹ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
x✝ : Invertible (NormedSpace.exp (z • star a)) := invertibleExp (z • star a)
⊢ x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z : ℂ
x✝¹ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
x✝ : Invertible (NormedSpace.exp (z • star a)) := invertibleExp (z • star a)
⊢ x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_2
inst✝ : NonUnitalCStarAlgebra A
a b x : A
ha : IsStarNormal a
hb : IsStarNormal b
h : SemiconjBy x a b
⊢ SemiconjBy ↑x ↑a ↑b

Tactic:
simpa using

Goal state after:
A : Type u_2
inst✝ : NonUnitalCStarAlgebra A
a b x : A
ha : IsStarNormal a
hb : IsStarNormal b
h : SemiconjBy x a b
⊢ ↑x * ↑a = ↑b * ↑x
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
hy : y ∈ B.polar ↑m
x : E
hx : x ∈ m
r : 𝕜
hr : (B x) y ≠ 0
⊢ ‖r‖ ≤ ‖(B x) y‖⁻¹

Tactic:
← one_div,

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
hy : y ∈ B.polar ↑m
x : E
hx : x ∈ m
r : 𝕜
hr : (B x) y ≠ 0
⊢ ‖r‖ ≤ 1 / ‖(B x) y‖
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
hy : y ∈ B.polar ↑m
x : E
hx : x ∈ m
r : 𝕜
hr : (B x) y ≠ 0
⊢ ‖r‖ * ‖(B x) y‖ ≤ 1

Tactic:
simpa using

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
hy : y ∈ B.polar ↑m
x : E
hx : x ∈ m
r : 𝕜
hr : (B x) y ≠ 0
⊢ ‖r‖ * ‖(B x) y‖ ≤ 1
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

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ Continuous fun p ↦ p.1 ⊔ p.2

Tactic:
isEmbedding_toCompacts.continuous_iff,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ Continuous (toCompacts ∘ fun p ↦ p.1 ⊔ p.2)
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:
isEmbedding_toCompacts.continuous_iff,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
⊢ Continuous (toCompacts ∘ fun p ↦ p.1 ×ˢ p.2)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → NonemptyCompacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
⊢ Continuous[inst✝², _] fun x ↦ NonemptyCompacts.map (g x) ⋯ (f x)

Tactic:
isEmbedding_toCompacts.continuous_iff,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → NonemptyCompacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
⊢ Continuous[inst✝², _] (toCompacts ∘ fun x ↦ NonemptyCompacts.map (g x) ⋯ (f x))
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ SeparableSpace (NonemptyCompacts α) ↔ SeparableSpace α

Tactic:
by
  refine ⟨fun _ => ?_, fun _ => inferInstance⟩
  rw [← Compacts.separableSpace_iff, ← isSeparable_univ_iff, ← union_compl_self {⊥},
    ← range_toCompacts]
  exact (finite_singleton _).isSeparable.union (isSeparable_range continuous_toCompacts)

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
K L : NonemptyCompacts α
hL : IsPreconnected ↑L
⊢ (range ⇑toCompactsOrderEmbedding).OrdConnected

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
K L : NonemptyCompacts α
hL : IsPreconnected ↑L
⊢ (Ioi ⊥).OrdConnected
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
K L : NonemptyCompacts α
hL : IsPreconnected ↑L
⊢ IsPreconnected (Ioc K L)

Tactic:
by
  rw [← isEmbedding_toCompacts.isPreconnected_image, ← coe_toCompactsOrderEmbedding,
    OrderEmbedding.image_Ioc _ (by simpa [← Set.Ioi_bot] using ordConnected_Ioi)]
  exact Compacts.isPreconnected_Ioc hL

Goal state after:
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✝ : PreconnectedSpace α
⊢ IsPreconnected univ

Tactic:
simpa using

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✝ : PreconnectedSpace α
⊢ IsPreconnected univ
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : Subspace 𝕜 E
f : StrongDual 𝕜 ↥p
g : StrongDual 𝕜 E
hg : ∀ (x : ↥p), g ↑x = ↑f x
hl : ∀ (x : E), ‖g x‖ ≤ (‖f‖₊ • normSeminorm 𝕜 E) x
x : ↥p
⊢ ‖f x‖ ≤ ‖g‖ * ‖x‖

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : Subspace 𝕜 E
f : StrongDual 𝕜 ↥p
g : StrongDual 𝕜 E
hg : ∀ (x : ↥p), g ↑x = ↑f x
hl : ∀ (x : E), ‖g x‖ ≤ (‖f‖₊ • normSeminorm 𝕜 E) x
x : ↥p
⊢ ‖f x‖ ≤ ‖g‖ * ‖↑x‖
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 : ∀ (i : ι), (f i) x = 0
this : Inseparable x 0
⊢ φ x = 0

Tactic:
simpa using

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), (f i) x = 0
this : Inseparable x 0
⊢ φ x = 0
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module ℝ E
inst✝ : PolynormableSpace ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp_cont : Continuous[inst✝², _] ⇑p
hp : ∀ (x : ↥S), f x ≤ p ↑x
g : Dual ℝ E
hg : ∀ (x : ↥S), g ↑x = f x
hl : ∀ (x : E), |g x| ≤ p x
x : E
⊢ g x ≤ (1 • {⟨p, hp_cont⟩}.sup fun p ↦ ↑p) x

Tactic:
simpa using

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module ℝ E
inst✝ : PolynormableSpace ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp_cont : Continuous[inst✝², _] ⇑p
hp : ∀ (x : ↥S), f x ≤ p ↑x
g : Dual ℝ E
hg : ∀ (x : ↥S), g ↑x = f x
hl : ∀ (x : E), |g x| ≤ p x
x : E
⊢ g x ≤ p x
Module: Mathlib.Analysis.Normed.Module.RCLike.Extend

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Module ℝ E
inst✝ : IsScalarTower ℝ 𝕜 E
fr : Dual ℝ E
p : Seminorm 𝕜 E
hp : ∀ (x : E), |fr x| ≤ p x
x : E
hx : ¬fr.extendRCLike x = 0
hsq : ‖fr.extendRCLike x‖ ^ 2 ≤ ‖fr.extendRCLike x‖ * p x
⊢ ‖fr.extendRCLike x‖ * ‖fr.extendRCLike x‖ ≤ p x * ‖fr.extendRCLike x‖

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Module ℝ E
inst✝ : IsScalarTower ℝ 𝕜 E
fr : Dual ℝ E
p : Seminorm 𝕜 E
hp : ∀ (x : E), |fr x| ≤ p x
x : E
hx : ¬fr.extendRCLike x = 0
hsq : ‖fr.extendRCLike x‖ ^ 2 ≤ ‖fr.extendRCLike x‖ * p x
⊢ ‖fr.extendRCLike x‖ * ‖fr.extendRCLike x‖ ≤ ‖fr.extendRCLike x‖ * p x
Module: Mathlib.Analysis.Normed.Operator.BanachSteinhaus

Goal state before:
pp: E : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_5
inst✝ : CompleteSpace E
g : ι → E →SL[σ₁₂] F
h : ∀ (x : E), ∃ C, ∀ (i : ι), ‖(g i) x‖ ≤ C
x✝ : Fin 1
x : E
⊢ BddAbove (range fun i ↦ (normSeminorm 𝕜₂ F) ((g i) x))

Tactic:
simpa using

Goal state after:
E : Type u_1
F : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
ι : Type u_5
inst✝ : CompleteSpace E
g : ι → E →SL[σ₁₂] F
h : ∀ (x : E), ∃ C, ∀ (i : ι), ‖(g i) x‖ ≤ C
x✝ : Fin 1
x : E
⊢ ∃ x_1, ∀ (a : ι), ‖(g a) x‖ ≤ x_1