File: Mathlib/Analysis/Convex/Independent.lean

full_name: ConvexIndependent
code: /-- An indexed family is said to be convex independent if every point only belongs to convex hulls
of sets containing it. -/
def ConvexIndependent (p : ι → E) : Prop 
goal: Prop

full_name: Subsingleton.convexIndependent
code: /-- A family with at most one point is convex independent. -/
theorem Subsingleton.convexIndependent [Subsingleton ι] (p : ι → E) : ConvexIndependent 𝕜 p 
goal: ConvexIndependent 𝕜 p

full_name: ConvexIndependent.injective
code: /-- A convex independent family is injective. -/
protected theorem ConvexIndependent.injective {p : ι → E} (hc : ConvexIndependent 𝕜 p) :
    Function.Injective p 
goal: Function.Injective p

full_name: ConvexIndependent.comp_embedding
code: /-- If a family is convex independent, so is any subfamily given by composition of an embedding into
index type with the original family. -/
theorem ConvexIndependent.comp_embedding {ι' : Type*} (f : ι' ↪ ι) {p : ι → E}
    (hc : ConvexIndependent 𝕜 p) : ConvexIndependent 𝕜 (p ∘ f) 
goal: ConvexIndependent 𝕜 (p ∘ ⇑f)

full_name: ConvexIndependent.subtype
code: /-- If a family is convex independent, so is any subfamily indexed by a subtype of the index type.
-/
protected theorem ConvexIndependent.subtype {p : ι → E} (hc : ConvexIndependent 𝕜 p) (s : Set ι) :
    ConvexIndependent 𝕜 fun i : s => p i 
goal: ConvexIndependent 𝕜 fun i => p ↑i

full_name: ConvexIndependent.range
code: /-- If an indexed family of points is convex independent, so is the corresponding set of points. -/
protected theorem ConvexIndependent.range {p : ι → E} (hc : ConvexIndependent 𝕜 p) :
    ConvexIndependent 𝕜 ((↑) : Set.range p → E) 
goal: ConvexIndependent 𝕜 Subtype.val

full_name: ConvexIndependent.mono
code: /-- A subset of a convex independent set of points is convex independent as well. -/
protected theorem ConvexIndependent.mono {s t : Set E} (hc : ConvexIndependent 𝕜 ((↑) : t → E))
    (hs : s ⊆ t) : ConvexIndependent 𝕜 ((↑) : s → E) 
goal: ConvexIndependent 𝕜 Subtype.val

full_name: Function.Injective.convexIndependent_iff_set
code: /-- The range of an injective indexed family of points is convex independent iff that family is. -/
theorem Function.Injective.convexIndependent_iff_set {p : ι → E} (hi : Function.Injective p) :
    ConvexIndependent 𝕜 ((↑) : Set.range p → E) ↔ ConvexIndependent 𝕜 p 
goal: ConvexIndependent 𝕜 Subtype.val ↔ ConvexIndependent 𝕜 p

full_name: ConvexIndependent.mem_convexHull_iff
code: /-- If a family is convex independent, a point in the family is in the convex hull of some of the
points given by a subset of the index type if and only if the point's index is in this subset. -/
@[simp]
protected theorem ConvexIndependent.mem_convexHull_iff {p : ι → E} (hc : ConvexIndependent 𝕜 p)
    (s : Set ι) (i : ι) : p i ∈ convexHull 𝕜 (p '' s) ↔ i ∈ s 
goal: p i ∈ (convexHull 𝕜) (p '' s) ↔ i ∈ s

full_name: convexIndependent_iff_not_mem_convexHull_diff
code: /-- If a family is convex independent, a point in the family is not in the convex hull of the other
points. See `convexIndependent_set_iff_not_mem_convexHull_diff` for the `Set` version. -/
theorem convexIndependent_iff_not_mem_convexHull_diff {p : ι → E} :
    ConvexIndependent 𝕜 p ↔ ∀ i s, p i ∉ convexHull 𝕜 (p '' (s \ {i})) 
goal: ConvexIndependent 𝕜 p ↔ ∀ (i : ι) (s : Set ι), p i ∉ (convexHull 𝕜) (p '' (s \ {i}))

full_name: convexIndependent_set_iff_inter_convexHull_subset
code: theorem convexIndependent_set_iff_inter_convexHull_subset {s : Set E} :
    ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ t, t ⊆ s → s ∩ convexHull 𝕜 t ⊆ t 
goal: ConvexIndependent 𝕜 Subtype.val ↔ ∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t

full_name: convexIndependent_set_iff_not_mem_convexHull_diff
code: /-- If a set is convex independent, a point in the set is not in the convex hull of the other
points. See `convexIndependent_iff_not_mem_convexHull_diff` for the indexed family version. -/
theorem convexIndependent_set_iff_not_mem_convexHull_diff {s : Set E} :
    ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ x ∈ s, x ∉ convexHull 𝕜 (s \ {x}) 
goal: ConvexIndependent 𝕜 Subtype.val ↔ ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})

full_name: convexIndependent_iff_finset
code: /-- To check convex independence, one only has to check finsets thanks to Carathéodory's theorem. -/
theorem convexIndependent_iff_finset {p : ι → E} :
    ConvexIndependent 𝕜 p ↔
      ∀ (s : Finset ι) (x : ι), p x ∈ convexHull 𝕜 (s.image p : Set E) → x ∈ s 
goal: ConvexIndependent 𝕜 p ↔ ∀ (s : Finset ι) (x : ι), p x ∈ (convexHull 𝕜) ↑(Finset.image p s) → x ∈ s

full_name: Convex.convexIndependent_extremePoints
code: theorem Convex.convexIndependent_extremePoints (hs : Convex 𝕜 s) :
    ConvexIndependent 𝕜 ((↑) : s.extremePoints 𝕜 → E) 
goal: ConvexIndependent 𝕜 Subtype.val