| import OAI.Computability.BinPacking.Inventory.InventoryPrefixConservation | |
| namespace OAI | |
| noncomputable section | |
| namespace BinPackingGap.InventoryData | |
| open scoped BigOperators | |
| variable (D : InventoryData) | |
| def packingOfRoleEquivs {Bin : Type*} [Fintype Bin] | |
| (e : ∀ r : Role, Bin ≃ D.RoleCopies r) | |
| (hcapacity : ∀ b, (∑ r : Role, D.itemSize ⟨r, e r b⟩) ≤ 1) : | |
| Packing D.packingInstance (Fintype.card Bin) := by | |
| classical | |
| let enum : Bin ≃ Fin (Fintype.card Bin) := Fintype.equivFin Bin | |
| let a : D.Item → Fin (Fintype.card Bin) := fun i => enum ((e i.1).symm i.2) | |
| refine packingOfFintype D.itemSize D.itemSize_pos D.itemSize_le_one a ?_ | |
| intro j | |
| change (∑ i : (Σ r : Role, D.RoleCopies r), | |
| if enum ((e i.1).symm i.2) = j then D.itemSize i else 0) ≤ 1 | |
| rw [Fintype.sum_sigma] | |
| have hsum : (∑ r : Role, ∑ c : D.RoleCopies r, | |
| if enum ((e r).symm c) = j then D.itemSize ⟨r, c⟩ else 0) = | |
| ∑ r : Role, D.itemSize ⟨r, e r (enum.symm j)⟩ := by | |
| apply Finset.sum_congr rfl | |
| intro r _ | |
| rw [← (e r).sum_comp] | |
| have he (b : Bin) : enum b = j ↔ b = enum.symm j := by | |
| constructor | |
| · intro h | |
| exact enum.injective (h.trans (enum.apply_symm_apply j).symm) | |
| · rintro rfl | |
| exact enum.apply_symm_apply j | |
| simp only [Equiv.symm_apply_apply, he] | |
| simp | |
| rw [hsum] | |
| exact hcapacity (enum.symm j) | |
| end BinPackingGap.InventoryData | |
| end | |
| end OAI | |