| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Basic |
|
|
| namespace OAI |
|
|
| /-! Parity-compatible triangular degrees and remainder bounds. -/ |
|
|
| noncomputable section |
| open scoped TensorProduct |
| namespace UniversalTensorSquares |
|
|
| def triangular (m : ℕ) : ℕ := m * (m + 1) / 2 |
|
|
| lemma triangular_eq_staircase_card (m : ℕ) : triangular m = (Saxl.staircase m).card := |
| (Saxl.staircase_card m).symm |
|
|
| @[simp] lemma triangular_zero : triangular 0 = 0 := rfl |
| @[simp] lemma triangular_one : triangular 1 = 1 := rfl |
|
|
| lemma triangular_succ (m : ℕ) : triangular (m + 1) = triangular m + m + 1 := by |
| simpa only [triangular_eq_staircase_card, Nat.add_assoc] using Saxl.staircase_card_succ m |
|
|
| lemma index_le_triangular (m : ℕ) : m ≤ triangular m := by |
| induction m with |
| | zero => simp |
| | succ m ih => rw [triangular_succ]; omega |
|
|
| lemma triangular_add_three (m : ℕ) : triangular (m + 3) = triangular m + 3 * m + 6 := by |
| have h1 := triangular_succ m |
| have h2 := triangular_succ (m + 1) |
| have h3 := triangular_succ (m + 2) |
| norm_num [Nat.add_assoc] at h1 h2 h3 |
| omega |
|
|
| def largestIndex (n : ℕ) : ℕ := |
| Nat.findGreatest (fun m => triangular m ≤ n ∧ triangular m % 2 = n % 2) n |
|
|
| lemma largestIndex_spec (n : ℕ) : |
| triangular (largestIndex n) ≤ n ∧ triangular (largestIndex n) % 2 = n % 2 := by |
| unfold largestIndex |
| rcases Nat.mod_two_eq_zero_or_one n with hn | hn |
| · apply Nat.findGreatest_spec (P := fun m => triangular m ≤ n ∧ triangular m % 2 = n % 2) (m := 0) (Nat.zero_le n) |
| exact ⟨by simp, by simpa using hn.symm⟩ |
| · have h1 : 1 ≤ n := by omega |
| apply Nat.findGreatest_spec (P := fun m => triangular m ≤ n ∧ triangular m % 2 = n % 2) (m := 1) h1 |
| exact ⟨by simpa using h1, by simpa using hn.symm⟩ |
|
|
| lemma largestIndex_maximal {n m : ℕ} (hmn : triangular m ≤ n) |
| (hp : triangular m % 2 = n % 2) : m ≤ largestIndex n := |
| Nat.le_findGreatest ((index_le_triangular m).trans hmn) ⟨hmn, hp⟩ |
|
|
| def remainderPairs (n : ℕ) : ℕ := (n - triangular (largestIndex n)) / 2 |
|
|
| lemma degree_decomposition (n : ℕ) : |
| triangular (largestIndex n) + 2 * remainderPairs n = n := by |
| have h := largestIndex_spec n |
| unfold remainderPairs |
| omega |
|
|
| lemma odd_index_remainder_bound (n : ℕ) (hodd : largestIndex n % 2 = 1) : |
| 2 * remainderPairs n ≤ largestIndex n - 1 := by |
| have hs := (largestIndex_spec n).2 |
| have ht := triangular_succ (largestIndex n) |
| have hp : triangular (largestIndex n + 1) % 2 = n % 2 := by omega |
| have hn : n < triangular (largestIndex n + 1) := by |
| by_contra h |
| have hmax := largestIndex_maximal (Nat.ge_of_not_lt h) hp |
| omega |
| have hd := degree_decomposition n |
| omega |
|
|
| lemma even_index_remainder_bound (n : ℕ) (heven : largestIndex n % 2 = 0) : |
| 2 * remainderPairs n ≤ 3 * largestIndex n + 4 := by |
| have hs := (largestIndex_spec n).2 |
| have ht := triangular_add_three (largestIndex n) |
| have hp : triangular (largestIndex n + 3) % 2 = n % 2 := by omega |
| have hn : n < triangular (largestIndex n + 3) := by |
| by_contra h |
| have hmax := largestIndex_maximal (Nat.ge_of_not_lt h) hp |
| omega |
| have hd := degree_decomposition n |
| omega |
|
|
| lemma small_index_degree_bound (n : ℕ) (hsmall : largestIndex n < 9) : n ≤ 64 := by |
| by_contra hn |
| rcases Nat.mod_two_eq_zero_or_one n with heven | hodd |
| · have h66 : 66 ≤ n := by omega |
| have hmax : 11 ≤ largestIndex n := largestIndex_maximal |
| (by norm_num [triangular]; exact h66) |
| (by norm_num [triangular]; exact heven.symm) |
| omega |
| · have h45 : 45 ≤ n := by omega |
| have hmax : 9 ≤ largestIndex n := largestIndex_maximal |
| (by norm_num [triangular]; exact h45) |
| (by norm_num [triangular]; exact hodd.symm) |
| omega |
| end UniversalTensorSquares |
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| end |
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| end OAI |
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