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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | import OAI.Probability.ContinuumTransition.TransformBounds
namespace OAI
universe u_1
section
open scoped Topology
open Set
namespace ContinuumTemperature
lemma isCompact_convexJoin {E : Type u_1} [AddCommGroup E] [Module ℝ E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
{S T : Set E} (hS : IsCompact S) (hT : IsCompact T) :
IsCompact (convexJoin ℝ S T) := by
have he : convexJoin ℝ S T =
(fun p : ℝ × E × E => (1-p.1) • p.2.1+p.1 • p.2.2) '' (Icc (0:ℝ) 1 ×ˢ S ×ˢ T) := by
ext z
simp only [mem_convexJoin,segment_eq_image,mem_image,mem_prod,Prod.exists]
constructor
· rintro ⟨x,hx,y,hy,c,hc,rfl⟩
exact ⟨c,x,y,⟨hc,hx,hy⟩,rfl⟩
· rintro ⟨c,x,y,⟨hc,hx,hy⟩,rfl⟩
exact ⟨x,hx,y,hy,c,hc,rfl⟩
rw [he]
exact (isCompact_Icc.prod (hS.prod hT)).image (by fun_prop)
lemma compact_convexHull_union {E : Type u_1} [AddCommGroup E] [Module ℝ E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
{S T : Set E} (hS : IsCompact S) (hT : IsCompact T)
(hcS : Convex ℝ S) (hcT : Convex ℝ T) : IsCompact (convexHull ℝ (S ∪ T)) := by
by_cases hs : S.Nonempty
· by_cases ht : T.Nonempty
· rw [hcS.convexHull_union hcT hs ht]
exact isCompact_convexJoin hS hT
· rw [not_nonempty_iff_eq_empty.mp ht,union_empty,hcS.convexHull_eq]
exact hS
· rw [not_nonempty_iff_eq_empty.mp hs,empty_union,hcT.convexHull_eq]
exact hT
lemma compact_convexHull_iUnion_fin {E : Type u_1} [AddCommGroup E] [Module ℝ E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
{n : ℕ} (K : Fin n → Set E) (hk : ∀ i, IsCompact (K i)) (hc : ∀ i, Convex ℝ (K i)) :
IsCompact (convexHull ℝ (⋃ i, K i)) := by
induction n with
| zero => simp
| succ n ih =>
rw [Set.iUnion_fin_add_one_eq_iUnion_succ,← convexHull_convexHull_union_right]
exact compact_convexHull_union (hk 0) (ih _ (fun i => hk i.succ) (fun i => hc i.succ))
(hc 0) (convex_convexHull _ _)
end ContinuumTemperature
end
end OAI
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