| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Support.WordContents |
|
|
| namespace OAI |
|
|
| /-! Part Words. -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Saxl |
| open scoped BigOperators |
|
|
| theorem exists_word_of_parts (ps : List ℕ) {n : ℕ} (hn : ps.sum = n) : |
| ∃ w : Fin n → Fin ps.length, ∀ j, wordContent w j = ps.get j := by |
| classical |
| have hc : Fintype.card (Σ j : Fin ps.length, Fin (ps.get j)) = n := by |
| rw [Fintype.card_sigma] |
| simp only [Fintype.card_fin] |
| rw [← List.sum_ofFn, List.ofFn_get, hn] |
| let e := Fintype.equivFinOfCardEq hc |
| refine ⟨fun i => (e.symm i).1, fun j => ?_⟩ |
| rw [wordContent,Finset.card_filter,← Equiv.sum_comp e,Fintype.sum_sigma] |
| simp |
|
|
| def balancedParts (n k : ℕ) : List ℕ := |
| List.ofFn (fun j : Fin k => n/k + if j.val < n%k then 1 else 0) |
|
|
| @[simp] lemma balancedParts_length (n k : ℕ) : (balancedParts n k).length = k := |
| List.length_ofFn |
|
|
| lemma balancedParts_part {n k p : ℕ} (hp : p ∈ balancedParts n k) : |
| n/k ≤ p ∧ p ≤ (n+k-1)/k := by |
| simp only [balancedParts, List.mem_ofFn] at hp |
| obtain ⟨j,rfl⟩ := hp |
| have hk : 0 < k := Nat.zero_lt_of_lt j.isLt |
| have hn : n = k*(n/k) + n%k := (Nat.div_add_mod n k).symm |
| have hr := Nat.mod_lt n hk |
| constructor |
| · omega |
| · split_ifs with h |
| · apply (Nat.le_div_iff_mul_le hk).mpr |
| have ht : n%k ≥ 1 := by omega |
| rw [Nat.add_mul,Nat.one_mul,Nat.mul_comm] |
| omega |
| · apply (Nat.le_div_iff_mul_le hk).mpr |
| simp only [Nat.add_zero] |
| rw [Nat.mul_comm] |
| omega |
|
|
| lemma balancedParts_pos {n k : ℕ} (hk : 0 < k) (hkn : k ≤ n) : |
| ∀ p ∈ balancedParts n k, 0 < p := by |
| intro p hp |
| have h := (balancedParts_part hp).1 |
| have hn : 1 ≤ n/k := (Nat.le_div_iff_mul_le hk).mpr (by simpa) |
| omega |
|
|
| lemma balancedParts_sum (n k : ℕ) (hk : 0 < k) : (balancedParts n k).sum = n := by |
| classical |
| rw [balancedParts,List.sum_ofFn,Finset.sum_add_distrib] |
| have hc : (∑ j : Fin k, if j.val < n%k then 1 else 0) = n%k := by |
| rw [← Finset.card_filter] |
| have he : Finset.univ.filter (fun j : Fin k => j.val < n%k) = |
| Finset.Iio ⟨n%k,Nat.mod_lt n hk⟩ := by |
| ext j |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_Iio,Fin.lt_def] |
| rw [he] |
| simp |
| rw [hc] |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| exact Nat.div_add_mod n k |
|
|
| end Saxl |
| end |
| end |
|
|
| end OAI |
|
|
