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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 | import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
import Mathlib.Basic.Real.Basic
import Mathlib.Tactic.NormNum
import OAI.Computability.BinPacking.Packing.PackingCoordinates
namespace OAI
noncomputable section
namespace BinPackingGap
def excludedPositions {T Pos : Type*} [Fintype Pos]
(selected : Finset T) (I : T → Pos → Prop) : Finset Pos := by
classical
exact Finset.univ.filter fun r => ∃ t ∈ selected, I t r
@[simp] theorem mem_excludedPositions {T Pos : Type*} [Fintype Pos]
(selected : Finset T) (I : T → Pos → Prop) {r : Pos} :
r ∈ excludedPositions selected I ↔ ∃ t ∈ selected, I t r := by
classical
simp [excludedPositions]
theorem excludedPositions_card_le {T Pos : Type*} [Fintype Pos]
(selected : Finset T) (I : T → Pos → Prop)
(hunique : ∀ t ∈ selected, ∀ r s, I t r → I t s → r = s) :
(excludedPositions selected I).card ≤ selected.card := by
classical
have witness (r : ↥(excludedPositions selected I)) : ∃ t ∈ selected, I t r :=
(mem_excludedPositions selected I).mp r.property
let f : ↥(excludedPositions selected I) → ↥selected := fun r =>
⟨(witness r).choose, (witness r).choose_spec.1⟩
have hf (r : ↥(excludedPositions selected I)) : I (f r) r :=
(witness r).choose_spec.2
have hinj : Function.Injective f := by
intro r s hrs
have ht : (f r : T) = (f s : T) := congrArg (fun t : ↥selected => (t : T)) hrs
apply Subtype.ext
apply hunique (f r) (f r).property r s (hf r)
rw [ht]
exact hf s
simpa only [Fintype.card_coe] using Fintype.card_le_of_injective f hinj
def excludedRepetitions {E : Type*} {R : ℕ}
(bad : Finset (E × Fin R)) (e : E) : Finset (Fin R) := by
classical
exact Finset.univ.filter fun j => (e, j) ∈ bad
def nonexcludedRepetitions {E : Type*} {R : ℕ}
(bad : Finset (E × Fin R)) (e : E) : Finset (Fin R) := by
classical
exact Finset.univ.filter fun j => (e, j) ∉ bad
@[simp] theorem mem_excludedRepetitions {E : Type*} {R : ℕ}
(bad : Finset (E × Fin R)) (e : E) {j : Fin R} :
j ∈ excludedRepetitions bad e ↔ (e, j) ∈ bad := by
classical
simp [excludedRepetitions]
@[simp] theorem mem_nonexcludedRepetitions {E : Type*} {R : ℕ}
(bad : Finset (E × Fin R)) (e : E) {j : Fin R} :
j ∈ nonexcludedRepetitions bad e ↔ (e, j) ∉ bad := by
classical
simp [nonexcludedRepetitions]
theorem excludedRepetitions_card_le {E : Type*} {R : ℕ}
(bad : Finset (E × Fin R)) (e : E) :
(excludedRepetitions bad e).card ≤ bad.card := by
classical
let f : ↥(excludedRepetitions bad e) → ↥bad := fun j =>
⟨(e, j), (mem_excludedRepetitions bad e).mp j.property⟩
have hinj : Function.Injective f := by
intro j k hjk
apply Subtype.ext
exact congrArg Prod.snd (congrArg (fun x : ↥bad => (x : E × Fin R)) hjk)
simpa only [Fintype.card_coe] using Fintype.card_le_of_injective f hinj
theorem excluded_nonexcluded_card {E : Type*} {R : ℕ}
(bad : Finset (E × Fin R)) (e : E) :
(excludedRepetitions bad e).card + (nonexcludedRepetitions bad e).card = R := by
classical
have hdis : Disjoint (excludedRepetitions bad e) (nonexcludedRepetitions bad e) := by
apply Finset.disjoint_left.mpr
intro j hj hk
exact ((mem_nonexcludedRepetitions bad e).mp hk)
((mem_excludedRepetitions bad e).mp hj)
have hunion : excludedRepetitions bad e ∪ nonexcludedRepetitions bad e = Finset.univ := by
ext j
by_cases hj : (e, j) ∈ bad <;> simp [hj]
calc
_ = (excludedRepetitions bad e ∪ nonexcludedRepetitions bad e).card :=
(Finset.card_union_of_disjoint hdis).symm
_ = R := by rw [hunion]; simp
theorem nonexcludedRepetitions_add_card_bound {E : Type*} {R P : ℕ}
(bad : Finset (E × Fin R)) (e : E) (hbad : bad.card ≤ P) :
R ≤ P + (nonexcludedRepetitions bad e).card := by
calc
R = (excludedRepetitions bad e).card + (nonexcludedRepetitions bad e).card :=
(excluded_nonexcluded_card bad e).symm
_ ≤ P + (nonexcludedRepetitions bad e).card :=
Nat.add_le_add_right ((excludedRepetitions_card_le bad e).trans hbad) _
theorem nonexcludedRepetitions_card_ge {E : Type*} {R P : ℕ}
(bad : Finset (E × Fin R)) (e : E) (hbad : bad.card ≤ P) :
R - P ≤ (nonexcludedRepetitions bad e).card := by
apply Nat.sub_le_iff_le_add.mpr
simpa only [Nat.add_comm] using nonexcludedRepetitions_add_card_bound bad e hbad
theorem nonexcludedRepetitions_card_ge_real {E : Type*} {R P : ℕ}
(bad : Finset (E × Fin R)) (e : E) (hbad : bad.card ≤ P) :
(R : ℝ) - P ≤ ((nonexcludedRepetitions bad e).card : ℝ) := by
have h : (R : ℝ) ≤ (P : ℝ) + ((nonexcludedRepetitions bad e).card : ℝ) := by
exact_mod_cast nonexcludedRepetitions_add_card_bound bad e hbad
apply sub_le_iff_le_add.mpr
simpa only [add_comm] using h
theorem nonexcludedRepetitions_card_ge_of_unique_hits {T E : Type*} [Fintype E]
{R P : ℕ} (selected : Finset T) (I : T → E × Fin R → Prop)
(hunique : ∀ t ∈ selected, ∀ r s, I t r → I t s → r = s)
(hselected : selected.card ≤ P) (e : E) :
R - P ≤ (nonexcludedRepetitions (excludedPositions selected I) e).card :=
nonexcludedRepetitions_card_ge _ e ((excludedPositions_card_le selected I hunique).trans hselected)
theorem nonexcludedRepetitions_card_ge_real_of_unique_hits {T E : Type*} [Fintype E]
{R P : ℕ} (selected : Finset T) (I : T → E × Fin R → Prop)
(hunique : ∀ t ∈ selected, ∀ r s, I t r → I t s → r = s)
(hselected : selected.card ≤ P) (e : E) :
(R : ℝ) - P ≤ ((nonexcludedRepetitions (excludedPositions selected I) e).card : ℝ) :=
nonexcludedRepetitions_card_ge_real _ e
((excludedPositions_card_le selected I hunique).trans hselected)
namespace PackingCounts
variable {D : InventoryData} {I : Instance} {b : ℕ}
def tupleHitsJob (p : Packing I b) (e : D.Item ≃ I.Item)
(t : p.TableBin (subclass e)) (r : D.Position) : Prop :=
(tupleClosedInterval p e t ∩
Geometry.closedJobCell D.graph.edges.length D.R D.geometryBound D.L r).Nonempty
def excludedAt (p : Packing I b) (e : D.Item ≃ I.Item) (v : D.Vertex) :
Finset D.Position :=
excludedPositions (zeroPositiveTuples p e v) (tupleHitsJob p e)
theorem selected_hits_job_unique (p : Packing I b) (e : D.Item ≃ I.Item)
(hL : 1 ≤ D.L) (v : D.Vertex) (t : p.TableBin (subclass e))
(ht : t ∈ zeroPositiveTuples p e v) {r s : D.Position}
(hr : tupleHitsJob p e t r) (hs : tupleHitsJob p e t s) : r = s :=
Geometry.closedJobCell_unique_of_short_interval hL
(selected_interval_length_first p e v t ht) hr hs
theorem excludedAt_card_le (p : Packing I b) (e : D.Item ≃ I.Item)
(hL : 1 ≤ D.L) (v : D.Vertex) : (excludedAt p e v).card ≤ D.P := by
apply (excludedPositions_card_le (zeroPositiveTuples p e v) (tupleHitsJob p e) ?_).trans
(zeroPositiveTuples_card_le p e v)
intro t ht r s hr hs
exact selected_hits_job_unique p e hL v t ht hr hs
def goodRepetitions (p : Packing I b) (e : D.Item ≃ I.Item)
(v : D.Vertex) (edge : D.Edge) : Finset (Fin D.R) :=
nonexcludedRepetitions (excludedAt p e v) edge
theorem goodRepetitions_card_ge (p : Packing I b) (e : D.Item ≃ I.Item)
(hL : 1 ≤ D.L) (v : D.Vertex) (edge : D.Edge) :
D.R - D.P ≤ (goodRepetitions p e v edge).card :=
nonexcludedRepetitions_card_ge _ edge (excludedAt_card_le p e hL v)
theorem goodRepetitions_card_ge_real (p : Packing I b) (e : D.Item ≃ I.Item)
(hL : 1 ≤ D.L) (v : D.Vertex) (edge : D.Edge) :
(D.R : ℝ) - D.P ≤ ((goodRepetitions p e v edge).card : ℝ) :=
nonexcludedRepetitions_card_ge_real _ edge (excludedAt_card_le p e hL v)
theorem covering_not_selected_of_nonexcluded (p : Packing I b) (e : D.Item ≃ I.Item)
(v : D.Vertex) (r : D.Position) (hr : r ∉ excludedAt p e v)
(t : p.TableBin (subclass e))
(ht : t ∈ tupleCovering p e v
(Geometry.jobInterior D.graph.edges.length D.R D.geometryBound D.L r)) :
t ∉ zeroPositiveTuples p e v := by
intro hselected
apply hr
apply (mem_excludedPositions _ _).mpr
refine ⟨t, hselected, ?_⟩
obtain ⟨_, hlo, hhi⟩ := (mem_tupleCovering p e v _ t).mp ht
refine ⟨Geometry.jobInterior D.graph.edges.length D.R D.geometryBound D.L r,
⟨hlo, hhi.le⟩, ?_⟩
exact Geometry.jobInterior_mem_closedJobCell _ _ _ _ _
end PackingCounts
end BinPackingGap
end
end OAI
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