File: Mathlib/Analysis/Convex/Intrinsic.lean

full_name: intrinsicInterior
code: /-- The intrinsic interior of a set is its interior considered as a set in its affine span. -/
def intrinsicInterior (s : Set P) : Set P 
goal: Set P

full_name: intrinsicFrontier
code: /-- The intrinsic frontier of a set is its frontier considered as a set in its affine span. -/
def intrinsicFrontier (s : Set P) : Set P 
goal: Set P

full_name: intrinsicClosure
code: /-- The intrinsic closure of a set is its closure considered as a set in its affine span. -/
def intrinsicClosure (s : Set P) : Set P 
goal: Set P

full_name: mem_intrinsicInterior
code: @[simp]
theorem mem_intrinsicInterior :
    x ∈ intrinsicInterior 𝕜 s ↔ ∃ y, y ∈ interior ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s) ∧ ↑y = x 
goal: x ∈ intrinsicInterior 𝕜 s ↔ ∃ y ∈ interior (Subtype.val ⁻¹' s), ↑y = x

full_name: mem_intrinsicFrontier
code: @[simp]
theorem mem_intrinsicFrontier :
    x ∈ intrinsicFrontier 𝕜 s ↔ ∃ y, y ∈ frontier ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s) ∧ ↑y = x 
goal: x ∈ intrinsicFrontier 𝕜 s ↔ ∃ y ∈ frontier (Subtype.val ⁻¹' s), ↑y = x

full_name: mem_intrinsicClosure
code: @[simp]
theorem mem_intrinsicClosure :
    x ∈ intrinsicClosure 𝕜 s ↔ ∃ y, y ∈ closure ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s) ∧ ↑y = x 
goal: x ∈ intrinsicClosure 𝕜 s ↔ ∃ y ∈ closure (Subtype.val ⁻¹' s), ↑y = x

full_name: intrinsicInterior_subset
code: theorem intrinsicInterior_subset : intrinsicInterior 𝕜 s ⊆ s 
goal: intrinsicInterior 𝕜 s ⊆ s

full_name: intrinsicFrontier_subset
code: theorem intrinsicFrontier_subset (hs : IsClosed s) : intrinsicFrontier 𝕜 s ⊆ s 
goal: intrinsicFrontier 𝕜 s ⊆ s

full_name: intrinsicFrontier_subset_intrinsicClosure
code: theorem intrinsicFrontier_subset_intrinsicClosure : intrinsicFrontier 𝕜 s ⊆ intrinsicClosure 𝕜 s 
goal: intrinsicFrontier 𝕜 s ⊆ intrinsicClosure 𝕜 s

full_name: subset_intrinsicClosure
code: theorem subset_intrinsicClosure : s ⊆ intrinsicClosure 𝕜 s 
goal: s ⊆ intrinsicClosure 𝕜 s

full_name: intrinsicInterior_empty
code: @[simp]
theorem intrinsicInterior_empty : intrinsicInterior 𝕜 (∅ : Set P) = ∅ 
goal: intrinsicInterior 𝕜 ∅ = ∅

full_name: intrinsicFrontier_empty
code: @[simp]
theorem intrinsicFrontier_empty : intrinsicFrontier 𝕜 (∅ : Set P) = ∅ 
goal: intrinsicFrontier 𝕜 ∅ = ∅

full_name: intrinsicClosure_empty
code: @[simp]
theorem intrinsicClosure_empty : intrinsicClosure 𝕜 (∅ : Set P) = ∅ 
goal: intrinsicClosure 𝕜 ∅ = ∅

full_name: intrinsicClosure_nonempty
code: @[simp]
theorem intrinsicClosure_nonempty : (intrinsicClosure 𝕜 s).Nonempty ↔ s.Nonempty 
goal: (intrinsicClosure 𝕜 s).Nonempty ↔ s.Nonempty

full_name: intrinsicInterior_singleton
code: @[simp]
theorem intrinsicInterior_singleton (x : P) : intrinsicInterior 𝕜 ({x} : Set P) = {x} 
goal: intrinsicInterior 𝕜 {x} = {x}

full_name: intrinsicFrontier_singleton
code: @[simp]
theorem intrinsicFrontier_singleton (x : P) : intrinsicFrontier 𝕜 ({x} : Set P) = ∅ 
goal: intrinsicFrontier 𝕜 {x} = ∅

full_name: intrinsicClosure_singleton
code: @[simp]
theorem intrinsicClosure_singleton (x : P) : intrinsicClosure 𝕜 ({x} : Set P) = {x} 
goal: intrinsicClosure 𝕜 {x} = {x}

full_name: intrinsicClosure_mono
code: theorem intrinsicClosure_mono (h : s ⊆ t) : intrinsicClosure 𝕜 s ⊆ intrinsicClosure 𝕜 t 
goal: intrinsicClosure 𝕜 s ⊆ intrinsicClosure 𝕜 t

full_name: interior_subset_intrinsicInterior
code: theorem interior_subset_intrinsicInterior : interior s ⊆ intrinsicInterior 𝕜 s 
goal: interior s ⊆ intrinsicInterior 𝕜 s

full_name: intrinsicClosure_subset_closure
code: theorem intrinsicClosure_subset_closure : intrinsicClosure 𝕜 s ⊆ closure s 
goal: intrinsicClosure 𝕜 s ⊆ closure s

full_name: intrinsicFrontier_subset_frontier
code: theorem intrinsicFrontier_subset_frontier : intrinsicFrontier 𝕜 s ⊆ frontier s 
goal: intrinsicFrontier 𝕜 s ⊆ frontier s

full_name: intrinsicClosure_subset_affineSpan
code: theorem intrinsicClosure_subset_affineSpan : intrinsicClosure 𝕜 s ⊆ affineSpan 𝕜 s 
goal: intrinsicClosure 𝕜 s ⊆ ↑(affineSpan 𝕜 s)

full_name: intrinsicClosure_diff_intrinsicFrontier
code: @[simp]
theorem intrinsicClosure_diff_intrinsicFrontier (s : Set P) :
    intrinsicClosure 𝕜 s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s 
goal: intrinsicClosure 𝕜 s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s

full_name: intrinsicClosure_diff_intrinsicInterior
code: @[simp]
theorem intrinsicClosure_diff_intrinsicInterior (s : Set P) :
    intrinsicClosure 𝕜 s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s 
goal: intrinsicClosure 𝕜 s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s

full_name: intrinsicInterior_union_intrinsicFrontier
code: @[simp]
theorem intrinsicInterior_union_intrinsicFrontier (s : Set P) :
    intrinsicInterior 𝕜 s ∪ intrinsicFrontier 𝕜 s = intrinsicClosure 𝕜 s 
goal: intrinsicInterior 𝕜 s ∪ intrinsicFrontier 𝕜 s = intrinsicClosure 𝕜 s

full_name: intrinsicFrontier_union_intrinsicInterior
code: @[simp]
theorem intrinsicFrontier_union_intrinsicInterior (s : Set P) :
    intrinsicFrontier 𝕜 s ∪ intrinsicInterior 𝕜 s = intrinsicClosure 𝕜 s 
goal: intrinsicFrontier 𝕜 s ∪ intrinsicInterior 𝕜 s = intrinsicClosure 𝕜 s

full_name: isClosed_intrinsicClosure
code: theorem isClosed_intrinsicClosure (hs : IsClosed (affineSpan 𝕜 s : Set P)) :
    IsClosed (intrinsicClosure 𝕜 s) 
goal: IsClosed (intrinsicClosure 𝕜 s)

full_name: isClosed_intrinsicFrontier
code: theorem isClosed_intrinsicFrontier (hs : IsClosed (affineSpan 𝕜 s : Set P)) :
    IsClosed (intrinsicFrontier 𝕜 s) 
goal: IsClosed (intrinsicFrontier 𝕜 s)

full_name: affineSpan_intrinsicClosure
code: @[simp]
theorem affineSpan_intrinsicClosure (s : Set P) :
    affineSpan 𝕜 (intrinsicClosure 𝕜 s) = affineSpan 𝕜 s 
goal: affineSpan 𝕜 (intrinsicClosure 𝕜 s) = affineSpan 𝕜 s

full_name: IsClosed.intrinsicClosure
code: protected theorem IsClosed.intrinsicClosure (hs : IsClosed ((↑) ⁻¹' s : Set <| affineSpan 𝕜 s)) :
    intrinsicClosure 𝕜 s = s 
goal: intrinsicClosure 𝕜 s = s

full_name: intrinsicClosure_idem
code: @[simp]
theorem intrinsicClosure_idem (s : Set P) :
    intrinsicClosure 𝕜 (intrinsicClosure 𝕜 s) = intrinsicClosure 𝕜 s 
goal: intrinsicClosure 𝕜 (intrinsicClosure 𝕜 s) = intrinsicClosure 𝕜 s

full_name: AffineIsometry.image_intrinsicInterior
code: @[simp]
theorem image_intrinsicInterior (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
    intrinsicInterior 𝕜 (φ '' s) = φ '' intrinsicInterior 𝕜 s 
goal: intrinsicInterior 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicInterior 𝕜 s

full_name: AffineIsometry.image_intrinsicFrontier
code: @[simp]
theorem image_intrinsicFrontier (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
    intrinsicFrontier 𝕜 (φ '' s) = φ '' intrinsicFrontier 𝕜 s 
goal: intrinsicFrontier 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicFrontier 𝕜 s

full_name: AffineIsometry.image_intrinsicClosure
code: @[simp]
theorem image_intrinsicClosure (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
    intrinsicClosure 𝕜 (φ '' s) = φ '' intrinsicClosure 𝕜 s 
goal: intrinsicClosure 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicClosure 𝕜 s

full_name: intrinsicClosure_eq_closure
code: @[simp]
theorem intrinsicClosure_eq_closure : intrinsicClosure 𝕜 s = closure s 
goal: intrinsicClosure 𝕜 s = closure s

full_name: closure_diff_intrinsicInterior
code: @[simp]
theorem closure_diff_intrinsicInterior (s : Set P) :
    closure s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s 
goal: closure s \ intrinsicInterior 𝕜 s = intrinsicFrontier 𝕜 s

full_name: closure_diff_intrinsicFrontier
code: @[simp]
theorem closure_diff_intrinsicFrontier (s : Set P) :
    closure s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s 
goal: closure s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s

full_name: _private.Mathlib.Analysis.Convex.Intrinsic.0.aux
code: private theorem aux {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] (φ : α ≃ₜ β)
    (s : Set β) : (interior s).Nonempty ↔ (interior (φ ⁻¹' s)).Nonempty 
goal: (interior s).Nonempty ↔ (interior (⇑φ ⁻¹' s)).Nonempty

full_name: Set.Nonempty.intrinsicInterior
code: /-- The intrinsic interior of a nonempty convex set is nonempty. -/
protected theorem Set.Nonempty.intrinsicInterior (hscv : Convex ℝ s) (hsne : s.Nonempty) :
    (intrinsicInterior ℝ s).Nonempty 
goal: (intrinsicInterior ℝ s).Nonempty

full_name: intrinsicInterior_nonempty
code: theorem intrinsicInterior_nonempty (hs : Convex ℝ s) :
    (intrinsicInterior ℝ s).Nonempty ↔ s.Nonempty 
goal: (intrinsicInterior ℝ s).Nonempty ↔ s.Nonempty