| import Lean.Elab.Tactic.Omega |
| import Mathlib.Algebra.BigOperators.Group.Finset.Basic |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Data.Fintype.Card |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Positivity |
|
|
| namespace OAI |
|
|
| namespace BinPackingGap.PrimaryLoss |
|
|
| open scoped BigOperators |
|
|
| variable {ι : Type*} [Fintype ι] {N : ℕ} |
|
|
| def binItems (f : ι → Fin N) (b : Fin N) : Finset ι := |
| Finset.univ.filter fun i => f i = b |
|
|
| abbrev fiber (f : ι → Fin N) (b : Fin N) : Finset ι := binItems f b |
|
|
| def itemsIn (f : ι → Fin N) (s : Finset (Fin N)) : Finset ι := |
| Finset.univ.filter fun i => f i ∈ s |
|
|
| def binPrimary (f : ι → Fin N) (p : ι → ℤ) (b : Fin N) : ℤ := |
| ∑ i ∈ binItems f b, p i |
|
|
| def fullBins (f : ι → Fin N) : Finset (Fin N) := |
| Finset.univ.filter fun b => (fiber f b).card = 5 |
|
|
| def nonfullBins (f : ι → Fin N) : Finset (Fin N) := |
| Finset.univ.filter fun b => (fiber f b).card < 5 |
|
|
| def nonfullItems (f : ι → Fin N) : Finset ι := itemsIn f (nonfullBins f) |
|
|
| def negativeFullBins (f : ι → Fin N) (p : ι → ℤ) : Finset (Fin N) := |
| (fullBins f).filter fun b => binPrimary f p b < 0 |
| |
| def outsideZeroFullItems (f : ι → Fin N) (p : ι → ℤ) : Finset ι := |
| Finset.univ.filter fun i => |
| ¬ ((fiber f (f i)).card = 5 ∧ binPrimary f p (f i) = 0) |
|
|
| theorem fibers_disjoint (f : ι → Fin N) {b d : Fin N} (h : b ≠ d) : |
| Disjoint (fiber f b) (fiber f d) := by |
| apply Finset.disjoint_left.mpr |
| intro i hi hd |
| simp only [fiber, binItems, Finset.mem_filter, Finset.mem_univ, true_and] at hi hd |
| exact h (hi.symm.trans hd) |
|
|
| theorem sum_fibers (f : ι → Fin N) (s : Finset (Fin N)) |
| {A : Type*} [AddCommMonoid A] (w : ι → A) : |
| (∑ b ∈ s, ∑ i ∈ fiber f b, w i) = ∑ i ∈ itemsIn f s, w i := by |
| exact Finset.sum_fiberwise_eq_sum_filter Finset.univ s f w |
|
|
| theorem card_itemsIn (f : ι → Fin N) (s : Finset (Fin N)) : |
| (itemsIn f s).card = ∑ b ∈ s, (fiber f b).card := by |
| exact (Finset.sum_card_fiberwise_eq_card_filter Finset.univ s f).symm |
|
|
| theorem sum_fiber_card (f : ι → Fin N) : |
| ∑ b, (fiber f b).card = Fintype.card ι := by |
| simpa [itemsIn] using (card_itemsIn f Finset.univ).symm |
|
|
| theorem sum_binPrimary (f : ι → Fin N) (p : ι → ℤ) : |
| ∑ b, binPrimary f p b = ∑ i, p i := by |
| simpa [itemsIn, binPrimary] using sum_fibers f Finset.univ p |
|
|
| theorem card_itemsIn_le (f : ι → Fin N) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) (s : Finset (Fin N)) : |
| (itemsIn f s).card ≤ 5 * s.card := by |
| rw [card_itemsIn] |
| calc |
| (∑ b ∈ s, (fiber f b).card) ≤ ∑ _ ∈ s, 5 := |
| Finset.sum_le_sum fun b _ => hcap b |
| _ = 5 * s.card := by simp [Nat.mul_comm] |
|
|
| theorem nonfullBins_card_le (f : ι → Fin N) (B c : ℕ) |
| (hcard : Fintype.card ι = 5 * B) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) (hN : N ≤ B + c) : |
| (nonfullBins f).card ≤ 5 * c := by |
| have hmass : ∑ b, (fiber f b).card = 5 * B := (sum_fiber_card f).trans hcard |
| have hdef : (∑ b, (5 - (fiber f b).card)) = 5 * N - 5 * B := by |
| rw [Finset.sum_tsub_distrib _ (fun b _ => hcap b), hmass] |
| simp [Nat.mul_comm] |
| calc |
| (nonfullBins f).card = ∑ _ ∈ nonfullBins f, 1 := by simp |
| _ ≤ ∑ b ∈ nonfullBins f, (5 - (fiber f b).card) := by |
| apply Finset.sum_le_sum |
| intro b hb |
| have hb' : (fiber f b).card < 5 := (Finset.mem_filter.mp hb).2 |
| omega |
| _ ≤ ∑ b : Fin N, (5 - (fiber f b).card) := |
| Finset.sum_le_sum_of_subset (Finset.subset_univ _) |
| _ ≤ 5 * c := by rw [hdef]; omega |
| |
| theorem nonfullItems_card_le (f : ι → Fin N) (B c : ℕ) |
| (hcard : Fintype.card ι = 5 * B) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) (hN : N ≤ B + c) : |
| (nonfullItems f).card ≤ 25 * c := by |
| have hi := card_itemsIn_le f hcap (nonfullBins f) |
| have hb := nonfullBins_card_le f B c hcard hcap hN |
| change (itemsIn f (nonfullBins f)).card ≤ 25 * c |
| omega |
| |
| theorem full_nonfull_disjoint (f : ι → Fin N) : |
| Disjoint (nonfullBins f) (fullBins f) := by |
| apply Finset.disjoint_left.mpr |
| intro b hn hf |
| have hn' := (Finset.mem_filter.mp hn).2 |
| have hf' := (Finset.mem_filter.mp hf).2 |
| omega |
| |
| theorem full_nonfull_union (f : ι → Fin N) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) : |
| nonfullBins f ∪ fullBins f = Finset.univ := by |
| ext b |
| simp only [Finset.mem_union, nonfullBins, fullBins, Finset.mem_filter, |
| Finset.mem_univ, true_and, iff_true] |
| have := hcap b |
| omega |
| |
| theorem primary_conservation (f : ι → Fin N) (p : ι → ℤ) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) (htotal : ∑ i, p i = 0) : |
| (∑ b ∈ nonfullBins f, binPrimary f p b) + |
| (∑ b ∈ fullBins f, binPrimary f p b) = 0 := by |
| rw [← Finset.sum_union (full_nonfull_disjoint f), full_nonfull_union f hcap, |
| sum_binPrimary, htotal] |
| |
| theorem negativeFullBins_card_le (f : ι → Fin N) (p : ι → ℤ) (B c P0 : ℕ) |
| (hcard : Fintype.card ι = 5 * B) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) (hN : N ≤ B + c) |
| (hp : ∀ i, p i ≤ (P0 : ℤ)) (htotal : ∑ i, p i = 0) |
| (hfull : ∀ b, (fiber f b).card = 5 → binPrimary f p b ≤ 0) : |
| (negativeFullBins f p).card ≤ 25 * c * P0 := by |
| have hnegative : ((negativeFullBins f p).card : ℤ) ≤ |
| -(∑ b ∈ fullBins f, binPrimary f p b) := by |
| calc |
| ((negativeFullBins f p).card : ℤ) = ∑ _ ∈ negativeFullBins f p, (1 : ℤ) := by simp |
| _ = ∑ b ∈ fullBins f, if binPrimary f p b < 0 then (1 : ℤ) else 0 := by |
| rw [negativeFullBins, Finset.sum_filter] |
| _ ≤ ∑ b ∈ fullBins f, -binPrimary f p b := by |
| apply Finset.sum_le_sum |
| intro b hb |
| have h := hfull b (Finset.mem_filter.mp hb).2 |
| split_ifs <;> omega |
| _ = _ := Finset.sum_neg_distrib (fun b => binPrimary f p b) |
| have hnonfull : (∑ b ∈ nonfullBins f, binPrimary f p b) ≤ |
| ((nonfullItems f).card : ℤ) * (P0 : ℤ) := by |
| change (∑ b ∈ nonfullBins f, ∑ i ∈ fiber f b, p i) ≤ _ |
| rw [sum_fibers] |
| calc |
| (∑ i ∈ itemsIn f (nonfullBins f), p i) ≤ |
| ∑ _ ∈ itemsIn f (nonfullBins f), (P0 : ℤ) := |
| Finset.sum_le_sum fun i _ => hp i |
| _ = _ := by simp [nonfullItems] |
| have hcons := primary_conservation f p hcap htotal |
| have hitems : ((nonfullItems f).card : ℤ) ≤ (25 * c : ℕ) := by |
| exact_mod_cast nonfullItems_card_le f B c hcard hcap hN |
| have hscaled := mul_le_mul_of_nonneg_right hitems (Int.natCast_nonneg P0) |
| have hbound : ((negativeFullBins f p).card : ℤ) ≤ ((25 * c * P0 : ℕ) : ℤ) := by |
| push_cast |
| push_cast at hscaled |
| linarith |
| exact_mod_cast hbound |
| |
| variable [DecidableEq ι] in |
| theorem outsideZeroFullItems_eq (f : ι → Fin N) (p : ι → ℤ) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) |
| (hfull : ∀ b, (fiber f b).card = 5 → binPrimary f p b ≤ 0) : |
| outsideZeroFullItems f p = nonfullItems f ∪ itemsIn f (negativeFullBins f p) := by |
| ext i |
| simp only [outsideZeroFullItems, nonfullItems, itemsIn, negativeFullBins, |
| nonfullBins, fullBins, Finset.mem_union, Finset.mem_filter, Finset.mem_univ, true_and] |
| have hc := hcap (f i) |
| have hs := hfull (f i) |
| omega |
| |
| theorem loss_items_disjoint (f : ι → Fin N) (p : ι → ℤ) : |
| Disjoint (nonfullItems f) (itemsIn f (negativeFullBins f p)) := by |
| apply Finset.disjoint_left.mpr |
| intro i hi hj |
| simp only [nonfullItems, itemsIn, negativeFullBins, nonfullBins, fullBins, |
| Finset.mem_filter, Finset.mem_univ, true_and] at hi hj |
| omega |
| |
| variable [DecidableEq ι] in |
| theorem primary_loss_bound (f : ι → Fin N) (p : ι → ℤ) (B c P0 : ℕ) |
| (hcard : Fintype.card ι = 5 * B) |
| (hcap : ∀ b, (fiber f b).card ≤ 5) (hN : N ≤ B + c) |
| (hp : ∀ i, |p i| < (P0 : ℤ)) (htotal : ∑ i, p i = 0) |
| (hfull : ∀ b, (fiber f b).card = 5 → binPrimary f p b ≤ 0) : |
| (nonfullItems f).card ≤ 25 * c ∧ |
| (negativeFullBins f p).card ≤ 25 * c * P0 ∧ |
| (outsideZeroFullItems f p).card ≤ 25 * c * (1 + 5 * P0) ∧ |
| (outsideZeroFullItems f p).card < 25 * (c + 1) * (1 + 5 * P0) := by |
| have hnf := nonfullItems_card_le f B c hcard hcap hN |
| have hneg := negativeFullBins_card_le f p B c P0 hcard hcap hN |
| (fun i => (le_abs_self (p i)).trans (hp i).le) htotal hfull |
| have hni := card_itemsIn_le f hcap (negativeFullBins f p) |
| have hbad : (outsideZeroFullItems f p).card ≤ 25 * c * (1 + 5 * P0) := by |
| rw [outsideZeroFullItems_eq f p hcap hfull, |
| Finset.card_union_of_disjoint (loss_items_disjoint f p)] |
| nlinarith |
| refine ⟨hnf, hneg, hbad, hbad.trans_lt ?_⟩ |
| nlinarith |
| |
| end BinPackingGap.PrimaryLoss |
| |
| end OAI |
| |
