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e3e0a0d | 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 | 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
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