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

Goal state before:
pp: A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x} ⊆
    (fun x ↦ cfcₙ x a) '' (fun x ↦ cfcₙ x a) ⁻¹' {a | 0 ≤ a}

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x} ⊆
    (fun x ↦ cfcₙ x a) '' (fun x ↦ cfcₙ x a) ⁻¹' {a | 0 ≤ a}
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
hmain : ‖M‖ ≤ √‖M * star M‖
⊢ √(‖M‖ * ‖M‖) ≤ √‖M * star M‖

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
hmain : ‖M‖ ≤ √‖M * star M‖
⊢ √(‖M‖ * ‖M‖) ≤ √‖M * star M‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' spectrum ℂ a)
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.re) '' spectrum ℂ a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (re '' spectrum ℂ a)
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.re) '' spectrum ℂ a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

Goal state before:
pp: case hf
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(quasispectrum 𝕜 a)
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ)

Tactic:
]

Goal state after:
case hf
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(quasispectrum 𝕜 a)
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x} =
    (fun x ↦ cfcₙ x a) '' (fun x ↦ cfcₙ x a) ⁻¹' {a | 0 ≤ a}

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x} =
    (fun x ↦ cfcₙ x a) '' (fun x ↦ cfcₙ x a) ⁻¹' {a | 0 ≤ a}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ → ℝ
hf : 0 ≤ cfcₙ f a
⊢ ?m.203

Tactic:
by
    simpa only [not_and_or] using
      em (ContinuousOn f (quasispectrum ℝ a) ∧ f 0 = 0 ∧ IsSelfAdjoint a)

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' spectrum ℂ a)
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' spectrum ℂ a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra ℂ A
inst✝³ : StarModule ℂ A
inst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMap.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (im '' spectrum ℂ a)
ha : IsStarNormal a
⊢ ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' spectrum ℂ a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: case inl
A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ → ℝ
hf : ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x
h₁ : ContinuousOn f (quasispectrum ℝ a)
h₂ : f 0 = 0
h₃ : IsSelfAdjoint a
⊢ ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x

Tactic:
]

Goal state after:
case inl
A : Type u_1
inst✝¹⁴ : NonUnitalRing A
inst✝¹³ : StarRing A
inst✝¹² : Module ℝ A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : SMulCommClass ℝ A A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : IsTopologicalRing A
inst✝⁷ : T2Space A
inst✝⁶ : PartialOrder A
inst✝⁵ : NonnegSpectrumClass ℝ A
inst✝⁴ : StarOrderedRing A
inst✝³ : StarModule ℝ A
inst✝² : ContinuousStar A
inst✝¹ : ContinuousConstSMul ℝ A
inst✝ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ → ℝ
hf : ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x
h₁ : ContinuousOn f (quasispectrum ℝ a)
h₂ : f 0 = 0
h₃ : IsSelfAdjoint a
⊢ ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ Set.EqOn (fun x ↦ star (f x) * f x) 1 (spectrum R a) ↔ ∀ x ∈ spectrum R a, star (f x) * f x = 1

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ Set.EqOn (fun x ↦ star (f x) * f x) 1 (spectrum R a) ↔ ∀ x ∈ spectrum R a, star (f x) * f x = 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary

Goal state before:
pp: A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
⊢ IsStarNormal u ∧ u ∈ unitary A ↔ IsStarNormal u ∧ spectrum ℂ u ⊆ ↑(unitary ℂ)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
⊢ IsStarNormal u ∧ u ∈ unitary A ↔ IsStarNormal u ∧ spectrum ℂ u ⊆ ↑(unitary ℂ)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary

Goal state before:
pp: A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
hu : IsStarNormal u
⊢ cfc id u ∈ unitary A ↔ spectrum ℂ u ⊆ ↑(unitary ℂ)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
hu : IsStarNormal u
⊢ cfc id u ∈ unitary A ↔ spectrum ℂ u ⊆ ↑(unitary ℂ)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary

Goal state before:
pp: A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
hu : IsStarNormal u
⊢ (∀ x ∈ spectrum ℂ u, star (id x) * id x = 1) ↔ ∀ x ∈ spectrum ℂ u, x ∈ ↑(unitary ℂ)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
hu : IsStarNormal u
⊢ (∀ x ∈ spectrum ℂ u, star (id x) * id x = 1) ↔ ∀ x ∈ spectrum ℂ u, x ∈ ↑(unitary ℂ)
Module: Mathlib.Analysis.Normed.Group.NullSubmodule

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedCommGroup M
x y : M
hx : ‖x‖ = 0
hy : ‖y‖ = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.NullSubmodule

Goal state before:
pp: M : Type u_1
inst✝⁴ : SeminormedCommGroup M
𝕜 : Type u_2
E : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
c : 𝕜
x : E
hx : ‖x‖ = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case e_6
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ :
  ∀ (z : ↑(quasispectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
key₂ :
  ∀ (z : ↑(quasispectrum 𝕜 a)),
    ∫ (x : X), f x ↑z ∂μ = (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z
z : ↑(quasispectrum 𝕜 a)
⊢ (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z =
    (mkD ((quasispectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
⊢ (starRingEnd 𝕜) ⟪b i, x⟫ = ⟪y, b i⟫

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
⊢ (starRingEnd 𝕜) ⟪b i, x⟫ = ⟪y, b i⟫
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
⊢ (starRingEnd 𝕜) ⟪x, b i⟫ = ⟪b i, y⟫

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪x, b i⟫ = ⟪y, b i⟫
i : ι
⊢ (starRingEnd 𝕜) ⟪x, b i⟫ = ⟪b i, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
htriv : ¬Yᗮ = ⊥
⊢ ∃ a, (toDualMap 𝕜 E) a = ℓ

Tactic:
]

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
htriv : ¬Yᗮ = ⊥
⊢ ∃ a, (toDualMap 𝕜 E) a = ℓ
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
htriv : ∃ x ∈ Yᗮ, x ≠ 0
⊢ ∃ a, (toDualMap 𝕜 E) a = ℓ

Tactic:
]

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
htriv : ∃ x ∈ Yᗮ, x ≠ 0
⊢ ∃ a, (toDualMap 𝕜 E) a = ℓ
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
⊢ ↑ℓ x * ℓ z - ℓ x * ↑ℓ z = 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
⊢ ↑ℓ x * ℓ z - ℓ x * ↑ℓ z = 0
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
⊢ 0 = 0

pp: case a
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
⊢ ℓ z • x - ℓ x • z ∈ Y

Tactic:
]

Goal state after:
case a
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
⊢ ℓ z • x - ℓ x • z ∈ Y
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
x : E
y : StrongDual 𝕜 E
⊢ ((toDual 𝕜 E) ((toDual 𝕜 E).symm y)) x = y x

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
x : E
y : StrongDual 𝕜 E
⊢ ((toDual 𝕜 E) ((toDual 𝕜 E).symm y)) x = y x
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
B : E →L⋆[𝕜] E →L[𝕜] 𝕜
v f : E
is_lax_milgram : ∀ (w : E), ⟪f, w⟫ = (B v) w
w : E
⊢ ⟪f, w⟫ = (B v) w

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
B : E →L⋆[𝕜] E →L[𝕜] 𝕜
v f : E
is_lax_milgram : ∀ (w : E), ⟪f, w⟫ = (B v) w
w : E
⊢ ⟪f, w⟫ = (B v) w
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ finrank 𝕜 ↥(𝕜 ∙ x) = 1

pp: case hf
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ (innerSL 𝕜) y ≠ 0

Tactic:
]

Goal state after:
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ finrank 𝕜 ↥(𝕜 ∙ x) = 1

case hf
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ (innerSL 𝕜) y ≠ 0
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A B : Set E
x : E
h : IsExtreme 𝕜 A B
y z : E
hx : x ∈ A
hy : y ∈ A
hz : z ∈ B
hzxy : z ∈ openSegment 𝕜 x y
⊢ z ∈ openSegment 𝕜 x y

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A B : Set E
x : E
h : IsExtreme 𝕜 A B
y z : E
hx : x ∈ A
hy : y ∈ A
hz : z ∈ B
hzxy : z ∈ openSegment 𝕜 x y
⊢ z ∈ openSegment 𝕜 x y
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
A : Set E
ι : Sort u_6
inst✝ : Nonempty ι
F : ι → Set E
hAF : ∀ (i : ι), IsExtreme 𝕜 A (F i)
inhabited_h : Inhabited ι
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hxF : ∀ (i : ι), x ∈ F i
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ ∀ (i : ι), x₁ ∈ F i

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
A : Set E
ι : Sort u_6
inst✝ : Nonempty ι
F : ι → Set E
hAF : ∀ (i : ι), IsExtreme 𝕜 A (F i)
inhabited_h : Inhabited ι
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hxF : ∀ (i : ι), x ∈ F i
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ ∀ (i : ι), x₁ ∈ F i
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
F : Set (Set E)
hF : F.Nonempty
hA : ∀ B ∈ F, IsExtreme 𝕜 A B
⊢ IsExtreme 𝕜 A (⋂ B ∈ F, B)

Tactic:
have := hF.to_subtype

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
F : Set (Set E)
hF : F.Nonempty
hA : ∀ B ∈ F, IsExtreme 𝕜 A B
this : Nonempty ↑F
⊢ IsExtreme 𝕜 A (⋂ B ∈ F, B)
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
x : E
h : x ∈ extremePoints 𝕜 A
x₁ : E
hx₁ : x₁ ∈ A
x₂ : E
hx₂ : x₂ ∈ A
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x ∈ openSegment 𝕜 x₁ x₂

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
x : E
h : x ∈ extremePoints 𝕜 A
x₁ : E
hx₁ : x₁ ∈ A
x₂ : E
hx₂ : x₂ ∈ A
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x ∈ openSegment 𝕜 x₁ x₂
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
t : Set F
x : E
y : F
hx : (x, y) ∈ s ×ˢ t
h : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × F⦄, x₂ ∈ s ×ˢ t → (x, y) ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y)
x₁ : E
hx₁ : x₁ ∈ s
x₂ : E
hx₂ : x₂ ∈ s
hx_fst : (x, y).1 ∈ openSegment 𝕜 x₁ x₂
⊢ (x, y) ∈ (fun x_1 ↦ (x_1, (x, y).2)) '' openSegment 𝕜 x₁ x₂

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : Semiring 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s : Set E
t : Set F
x : E
y : F
hx : (x, y) ∈ s ×ˢ t
h : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × F⦄, x₂ ∈ s ×ˢ t → (x, y) ∈ openSegment 𝕜 x₁ x₂ → x₁ = (x, y)
x₁ : E
hx₁ : x₁ ∈ s
x₂ : E
hx₂ : x₂ ∈ s
hx_fst : (x, y).1 ∈ openSegment 𝕜 x₁ x₂
⊢ (x, y) ∈ (fun x_1 ↦ (x_1, (x, y).2)) '' openSegment 𝕜 x₁ x₂