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8df23f4 | 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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.Computability.BinPacking.Inventory.InventoryPrefixConservation
import OAI.Computability.BinPacking.Inventory.LocalCompletionCoverage
import OAI.Computability.BinPacking.Machines.PackingJobCoverage
import OAI.Computability.BinPacking.Packing.PackingFeasibility
namespace OAI
noncomputable section
namespace BinPackingGap
namespace InventoryCoverage
variable {D : InventoryData} {I : Instance} {b : ℕ}
theorem coveringTuplesAt_eq (p : Packing I b) (e : D.Item ≃ I.Item)
(v : D.Vertex) (a : ℚ) :
Coverage.coveringTuplesAt p (PackingCounts.subclass e) (PackingCounts.label e) v
(rowBaseline e) (PackingCounts.tupleFinish p e) a =
PackingCounts.tupleCovering p e v a := by
unfold Coverage.coveringTuplesAt PackingCounts.tupleCovering
apply Finset.filter_congr
intro t _
rw [rowBaseline_selected p e t]
rfl
theorem shortMainAt_eq (p : Packing I b) (e : D.Item ≃ I.Item)
(v : D.Vertex) (r : D.Position) :
Coverage.currentShortMainAt p (PackingCounts.subclass e) (PackingCounts.label e) v
(rowBaseline e) (rowShort e) (Geometry.baseline D.graph.edges.length D.R r) =
PackingCounts.shortMainAt p e v r := by
unfold Coverage.currentShortMainAt PackingCounts.shortMainAt
apply Finset.filter_congr
intro t _
rw [rowBaseline_selected p e t, rowShort_selected p e t]
rfl
end InventoryCoverage
namespace InventoryData
variable (D : InventoryData)
theorem test_coverage {b c : ℕ}
(hplusheight : D.plus.height = D.L) (hminusheight : D.minus.height = D.L)
(hL : 1 ≤ D.L) (hk : D.k ≤ D.graph.n)
(p : Packing D.packingInstance b) (hb : b ≤ D.B + c)
(v : D.Vertex) (t : D.TestAt v) (a : ℚ) (ha : a ∈ D.testInterval t) :
D.d ≤ (PackingCounts.tupleCovering p D.itemEquiv v a).card + 6 * lossAllowance c := by
classical
have hbad := D.badAnchorsAt_card_le hk p hb v
have horder := PackingCounts.edge_selected_baseline_le p D.itemEquiv
D.packingInstance_size_eq v
have hle : ∀ u ∈ Coverage.mainTuplesAt p (PackingCounts.subclass D.itemEquiv) (PackingCounts.label D.itemEquiv) v,
PackingCounts.tupleFinish p D.itemEquiv u ≤
InventoryCoverage.rowBaseline D.itemEquiv (p.itemAtRole (PackingCounts.subclass D.itemEquiv) u .x) := by
intro u _
simpa only [InventoryCoverage.rowBaseline_selected, PackingCounts.tupleBaseline] using
PackingCounts.tupleFinish_le_baseline p D.itemEquiv u
have hfeasible := PackingCounts.main_completion_le_selected_deadline p D.itemEquiv
D.packingInstance_size_eq v
have hdeficit := InventoryCoverage.actual_deadline_deficit_bound p D.itemEquiv v a
(lossAllowance c) (PackingCounts.tupleFinish p D.itemEquiv) hbad horder hle hfeasible
rw [D.deadline_prefix_at_test hplusheight hminusheight hL v t a ha,
InventoryCoverage.coveringTuplesAt_eq] at hdeficit
change (InventoryCoverage.originalBaselinePrefix D v a).card + D.d ≤
(InventoryCoverage.originalBaselinePrefix D v a).card +
(PackingCounts.tupleCovering p D.itemEquiv v a).card + 6 * lossAllowance c at hdeficit
omega
def longMatchesAt {b : ℕ} (p : Packing D.packingInstance b)
(v : D.Vertex) (r : D.IncidencePosition v) : ℕ :=
(Coverage.currentLongEdgeAt p D.encodedSubclass D.encodedLabel v
(InventoryCoverage.rowBaseline D.itemEquiv) (InventoryCoverage.keyBaseline D.itemEquiv)
(InventoryCoverage.rowShort D.itemEquiv)
(Geometry.baseline D.graph.edges.length D.R (D.incidencePosition r))).card
theorem long_matches_at_nonexcluded {b c : ℕ}
(hplusheight : D.plus.height = D.L) (hminusheight : D.minus.height = D.L)
(hL : 1 ≤ D.L) (hk : D.k ≤ D.graph.n)
(p : Packing D.packingInstance b) (hb : b ≤ D.B + c)
(v : D.Vertex) (r : D.IncidencePosition v)
(hr : D.incidencePosition r ∉ PackingCounts.excludedAt p D.itemEquiv v) :
D.d ≤ D.longMatchesAt p v r + PackingCounts.xplus p D.itemEquiv v + 9 * lossAllowance c := by
classical
let a := Geometry.jobInterior D.graph.edges.length D.R D.geometryBound D.L (D.incidencePosition r)
let finish := PackingCounts.tupleFinish p D.itemEquiv
have hbad := D.badAnchorsAt_card_le hk p hb v
have horder := PackingCounts.edge_selected_baseline_le p D.itemEquiv
D.packingInstance_size_eq v
have hle : ∀ u ∈ Coverage.mainTuplesAt p (PackingCounts.subclass D.itemEquiv) (PackingCounts.label D.itemEquiv) v,
finish u ≤ InventoryCoverage.rowBaseline D.itemEquiv
(p.itemAtRole (PackingCounts.subclass D.itemEquiv) u .x) := by
intro u _
simpa only [finish, InventoryCoverage.rowBaseline_selected, PackingCounts.tupleBaseline] using
PackingCounts.tupleFinish_le_baseline p D.itemEquiv u
have hfeasible := PackingCounts.main_completion_le_selected_deadline p D.itemEquiv
D.packingInstance_size_eq v
have hpartition := Coverage.completion_prefix_decomposition p (PackingCounts.subclass D.itemEquiv)
(PackingCounts.label D.itemEquiv) v (InventoryCoverage.rowBaseline D.itemEquiv) finish hle a
have hdeadline := Coverage.deadline_prefix_le_completion_add_loss p (PackingCounts.subclass D.itemEquiv)
(PackingCounts.label D.itemEquiv) v (InventoryCoverage.anchorDeadline D.itemEquiv) finish a
(lossAllowance c) hbad hfeasible
have ha : a ∈ D.testInterval (.inr (.inr r) : D.TestAt v) :=
Geometry.jobInterior_mem_jobTestInterval D.graph.edges.length D.R D.geometryBound D.L
(D.incidencePosition r)
rw [InventoryCoverage.deadlinePrefix_card,
D.deadline_prefix_at_test hplusheight hminusheight hL v (.inr (.inr r)) a ha] at hdeadline
have hbase := InventoryCoverage.actual_displaced_job_prefix_le p D.itemEquiv v r
(lossAllowance c) hbad horder
have hupper : (Coverage.coveringTuplesAt p (PackingCounts.subclass D.itemEquiv) (PackingCounts.label D.itemEquiv) v
(InventoryCoverage.rowBaseline D.itemEquiv) finish a).card ≤
PackingCounts.xplus p D.itemEquiv v +
(Coverage.currentShortMainAt p (PackingCounts.subclass D.itemEquiv) (PackingCounts.label D.itemEquiv) v
(InventoryCoverage.rowBaseline D.itemEquiv) (InventoryCoverage.rowShort D.itemEquiv)
(Geometry.baseline D.graph.edges.length D.R (D.incidencePosition r))).card := by
simpa only [finish, a, InventoryCoverage.coveringTuplesAt_eq,
InventoryCoverage.shortMainAt_eq] using
PackingCounts.nonexcluded_coverage_upper p D.itemEquiv v (D.incidencePosition r) hr
have hcorrection := CoverageCounts.displaced_short_correction hpartition hdeadline hbase hupper
exact InventoryCoverage.actual_current_long_matches_from_correction p D.itemEquiv v r
(lossAllowance c) (PackingCounts.xplus p D.itemEquiv v) hbad horder hcorrection
end InventoryData
end BinPackingGap
end
end OAI
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