| import OAI.Computability.BinPacking.Packing.PackingCoordinates | |
| namespace OAI | |
| namespace BinPackingGap.InventoryData | |
| variable (D : InventoryData) | |
| private theorem node_right_in_root {T : UniformTree} (v : T.Node) | |
| (s : TreeGeometry.Side) (hT : T.branchBound ≤ D.geometryBound) : | |
| TreeGeometry.rootOffset s ≤ | |
| TreeGeometry.right D.graph.edges.length D.R D.geometryBound s v.val ∧ | |
| TreeGeometry.right D.graph.edges.length D.R D.geometryBound s v.val ≤ | |
| TreeGeometry.rootOffset s + 1 := by | |
| apply TreeGeometry.nodeCell_subset_root D.graph.edges.length D.R D.geometryBound s hT v | |
| exact ⟨(TreeGeometry.left_lt_right _ _ _ _ _).le, le_rfl⟩ | |
| theorem rowBaseline_le_one_of_not_tm {v : D.Vertex} (r : D.RowAt v) | |
| (hclass : D.rowSubclass r ≠ .tm) : D.rowBaseline r ≤ 1 := by | |
| rcases r with t | ⟨short, j⟩ | |
| · rcases t with ⟨node, i⟩ | ⟨node, i⟩ | i | |
| · have h := D.node_right_in_root node .plus D.plus_branch_le_geometryBound | |
| simpa only [rowBaseline, treeBaseline, TreeGeometry.rootOffset, zero_add] using h.2 | |
| · exact (hclass rfl).elim | |
| · exact le_rfl | |
| · exact (Geometry.job_baseline_lt_one _ _ _).le | |
| theorem rowBaseline_ge_four_of_tm {v : D.Vertex} (r : D.RowAt v) | |
| (hclass : D.rowSubclass r = .tm) : 4 ≤ D.rowBaseline r := by | |
| rcases r with t | ⟨short, j⟩ | |
| · rcases t with ⟨node, i⟩ | ⟨node, i⟩ | i | |
| · cases hclass | |
| · exact (D.node_right_in_root node .minus D.minus_branch_le_geometryBound).1 | |
| · cases hclass | |
| · cases hclass | |
| theorem rowShort_eq_false_of_tm {v : D.Vertex} (r : D.RowAt v) | |
| (hclass : D.rowSubclass r = .tm) : D.rowShort r = false := by | |
| rcases r with t | ⟨short, j⟩ | |
| · rfl | |
| · cases hclass | |
| theorem exists_jobCopy_of_rowShort {v : D.Vertex} (r : D.RowAt v) | |
| (hshort : D.rowShort r = true) : ∃ j : D.JobCopy v, r = .inr (true, j) := by | |
| rcases r with t | ⟨short, j⟩ | |
| · cases hshort | |
| · change short = true at hshort | |
| subst short | |
| exact ⟨j, rfl⟩ | |
| theorem row_zero_resources_empty {v : D.Vertex} (r : D.RowAt v) | |
| (hshort : D.rowShort r = false) : | |
| completionInterval (D.rowBaseline r) (D.rowShort r) 0 0 | |
| (Geometry.delta D.graph.edges.length D.R D.geometryBound D.L) = ∅ := by | |
| rw [hshort] | |
| exact completionInterval_long_zero_lengths _ _ | |
| theorem row_zero_resources_job {v : D.Vertex} (r : D.RowAt v) | |
| (hshort : D.rowShort r = true) : | |
| ∃ j : D.JobCopy v, r = .inr (true, j) ∧ | |
| D.rowPosition r = some (D.jobPosition j) ∧ | |
| completionInterval (D.rowBaseline r) (D.rowShort r) 0 0 | |
| (Geometry.delta D.graph.edges.length D.R D.geometryBound D.L) = | |
| Geometry.jobTestInterval D.graph.edges.length D.R D.geometryBound D.L | |
| (D.jobPosition j) := by | |
| obtain ⟨j, rfl⟩ := D.exists_jobCopy_of_rowShort r hshort | |
| refine ⟨j, rfl, rfl, ?_⟩ | |
| exact completion_job_interval _ _ _ _ _ | |
| theorem minus_row_completion_gt_one {v w : D.Vertex} (r : D.RowAt v) | |
| (u : D.GlobalCopy) (l : D.LocalAt w) (hclass : D.rowSubclass r = .tm) : | |
| 1 < completion (D.rowBaseline r) (D.rowShort r) (D.globalLength u) | |
| (D.localLength l) (Geometry.delta D.graph.edges.length D.R D.geometryBound D.L) := by | |
| apply completion_minus_geometry_gt_one | |
| D.graph.edges.length D.R D.geometryBound D.L | |
| · exact D.rowBaseline_ge_four_of_tm r hclass | |
| · exact D.rowShort_eq_false_of_tm r hclass | |
| · exact D.globalLength_le_one u | |
| · exact D.localLength_le_lambda_one l | |
| theorem minus_row_not_cover_jobInterior {v w : D.Vertex} (r : D.RowAt v) | |
| (u : D.GlobalCopy) (l : D.LocalAt w) (hclass : D.rowSubclass r = .tm) | |
| (job : D.Position) : | |
| Geometry.jobInterior D.graph.edges.length D.R D.geometryBound D.L job ∉ | |
| completionInterval (D.rowBaseline r) (D.rowShort r) (D.globalLength u) | |
| (D.localLength l) (Geometry.delta D.graph.edges.length D.R D.geometryBound D.L) := by | |
| intro hmem | |
| have hh := D.minus_row_completion_gt_one r u l hclass | |
| have hj := (Geometry.closedJobCell_bounds | |
| (Geometry.jobInterior_mem_closedJobCell D.graph.edges.length D.R | |
| D.geometryBound D.L job)).2 | |
| exact (not_le_of_gt (hj.trans hh)) hmem.1 | |
| theorem plus_row_not_cover_minus_side {v w : D.Vertex} (r : D.RowAt v) | |
| (u : D.GlobalCopy) (l : D.LocalAt w) (hclass : D.rowSubclass r ≠ .tm) | |
| {a : ℚ} (ha : 4 ≤ a) : | |
| a ∉ completionInterval (D.rowBaseline r) (D.rowShort r) (D.globalLength u) | |
| (D.localLength l) (Geometry.delta D.graph.edges.length D.R D.geometryBound D.L) := by | |
| apply not_mem_plus_completionInterval | |
| · exact D.rowBaseline_le_one_of_not_tm r hclass | |
| · linarith | |
| end BinPackingGap.InventoryData | |
| end OAI | |