File size: 1,774 Bytes
8e4d533 | 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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | import Mathlib
import OAI.Geometry.Cannon.Modulus.Weights
namespace OAI
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open Set Topology
open Set Metric Topology
open Set Topology Filter Metric unitInterval
open scoped InnerProductSpace
namespace CannonCells
open Set
variable {X ι : Type*} [Fintype ι]
noncomputable def cost (C : ι → Set X) (w : ι → ℝ) (A : Set X) : ℝ :=
∑ i ∈ indices C A, w i
lemma cost_nonneg (C : ι → Set X) {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i) (A : Set X) :
0 ≤ cost C w A := Finset.sum_nonneg (fun i _ => hw i)
lemma indices_mono (C : ι → Set X) {A B : Set X} (h : A ⊆ B) : indices C A ⊆ indices C B := by
intro i hi
obtain ⟨x,hx,hxi⟩ := (mem_indices C A i).mp hi
exact (mem_indices C B i).mpr ⟨x,h hx,hxi⟩
lemma cost_mono (C : ι → Set X) {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i)
{A B : Set X} (h : A ⊆ B) : cost C w A ≤ cost C w B := by
classical
exact Finset.sum_le_sum_of_subset_of_nonneg (indices_mono C h) (fun i _ _ => hw i)
lemma indices_union [DecidableEq ι] (C : ι → Set X) (A B : Set X) :
indices C (A ∪ B) = indices C A ∪ indices C B := by
classical
ext i
simp only [mem_indices,Finset.mem_union,union_inter_distrib_right,union_nonempty]
lemma cost_union_le (C : ι → Set X) {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i) (A B : Set X) :
cost C w (A ∪ B) ≤ cost C w A + cost C w B := by
classical
unfold cost
rw [indices_union]
have h := Finset.sum_union_inter (s₁ := indices C A) (s₂ := indices C B) (f := w)
have hn : 0 ≤ ∑ i ∈ indices C A ∩ indices C B, w i :=
Finset.sum_nonneg (fun i _ => hw i)
linarith
end CannonCells
end
end
end
end
end
end
end
end
end
end OAI
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