| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Support.Extensions |
|
|
| namespace OAI |
|
|
| /-! Candidate. -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace UniversalTensorSquare |
|
|
| def rectangle (a b : ℕ) : YoungDiagram where |
| cells := Finset.range a ×ˢ Finset.range b |
| isLowerSet := by |
| intro x y hxy hy |
| simp only [Finset.mem_coe, Finset.mem_product, Finset.mem_range] at hy ⊢ |
| exact ⟨lt_of_le_of_lt hxy.1 hy.1, lt_of_le_of_lt hxy.2 hy.2⟩ |
|
|
| |
| (i, j) ∈ rectangle a b ↔ i < a ∧ j < b := by |
| change (i, j) ∈ Finset.range a ×ˢ Finset.range b ↔ _ |
| simp |
|
|
| def candidateArm (M b δ : ℕ) : YoungDiagram := |
| rectangle 1 (M + b + δ) ⊔ rectangle 2 (M - 1 + b) |
|
|
| def candidate (M b δ : ℕ) : YoungDiagram := |
| Saxl.staircase M ⊔ candidateArm M b δ ⊔ (candidateArm M b δ).transpose |
|
|
| lemma mem_candidate {M b δ i j : ℕ} : |
| (i, j) ∈ candidate M b δ ↔ |
| i + j < M ∨ |
| (i < 1 ∧ j < M + b + δ) ∨ (i < 2 ∧ j < M - 1 + b) ∨ |
| (j < 1 ∧ i < M + b + δ) ∨ (j < 2 ∧ i < M - 1 + b) := by |
| simp only [candidate, candidateArm, YoungDiagram.mem_sup, Saxl.mem_staircase, |
| mem_rectangle, YoungDiagram.mem_transpose, Prod.swap_prod_mk] |
| tauto |
|
|
| |
| (candidate M b δ).transpose = candidate M b δ := by |
| apply SetLike.ext |
| rintro ⟨i, j⟩ |
| simp only [YoungDiagram.mem_transpose, Prod.swap_prod_mk, mem_candidate] |
| omega |
|
|
| def armExtras (M b δ : ℕ) : Finset (ℕ × ℕ) := |
| {0} ×ˢ Finset.Ico M (M + b + δ) ∪ |
| {1} ×ˢ Finset.Ico (M - 1) (M - 1 + b) |
|
|
| lemma mem_armExtras {M b δ i j : ℕ} : |
| (i, j) ∈ armExtras M b δ ↔ |
| (i = 0 ∧ M ≤ j ∧ j < M + b + δ) ∨ |
| (i = 1 ∧ M - 1 ≤ j ∧ j < M - 1 + b) := by |
| simp only [armExtras, Finset.mem_union, Finset.mem_product, Finset.mem_singleton, |
| Finset.mem_Ico] |
|
|
| lemma armExtras_card (M b δ : ℕ) : (armExtras M b δ).card = 2 * b + δ := by |
| unfold armExtras |
| rw [Finset.card_union_of_disjoint] |
| · simp only [Finset.card_product, Finset.card_singleton, Nat.card_Ico, one_mul] |
| omega |
| · apply Finset.disjoint_left.mpr |
| rintro ⟨i, j⟩ hi hj |
| simp only [Finset.mem_product, Finset.mem_singleton] at hi hj |
| omega |
|
|
| lemma staircase_disjoint_armExtras (M b δ : ℕ) (hM : 1 ≤ M) : |
| Disjoint (Saxl.staircase M).cells (armExtras M b δ) := by |
| apply Finset.disjoint_left.mpr |
| rintro ⟨i, j⟩ hi hj |
| simp only [YoungDiagram.mem_cells, Saxl.mem_staircase] at hi |
| rw [mem_armExtras] at hj |
| omega |
|
|
| lemma armExtras_disjoint_transpose (M b δ : ℕ) (hM : 3 ≤ M) : |
| Disjoint (armExtras M b δ) |
| ((armExtras M b δ).map (Equiv.prodComm ℕ ℕ).toEmbedding) := by |
| apply Finset.disjoint_left.mpr |
| rintro ⟨i, j⟩ hi hj |
| simp only [Finset.mem_map_equiv, Equiv.prodComm_symm, Equiv.prodComm_apply, |
| Prod.swap_prod_mk] at hj |
| simp only [mem_armExtras] at hi hj |
| omega |
|
|
| lemma candidate_cells (M b δ : ℕ) (hM : 1 ≤ M) : |
| (candidate M b δ).cells = |
| (Saxl.staircase M).cells ∪ armExtras M b δ ∪ |
| (armExtras M b δ).map (Equiv.prodComm ℕ ℕ).toEmbedding := by |
| ext ⟨i, j⟩ |
| simp only [YoungDiagram.mem_cells, mem_candidate, Finset.mem_union, |
| Saxl.mem_staircase, mem_armExtras, Finset.mem_map_equiv, |
| Equiv.prodComm_symm, Equiv.prodComm_apply, |
| Prod.swap_prod_mk] |
| omega |
|
|
| lemma candidate_card (M b δ : ℕ) (hM : 3 ≤ M) : |
| (candidate M b δ).card = (Saxl.staircase M).card + 4 * b + 2 * δ := by |
| change (candidate M b δ).cells.card = _ |
| rw [candidate_cells M b δ (by omega)] |
| have hd : Disjoint (Saxl.staircase M).cells |
| ((armExtras M b δ).map (Equiv.prodComm ℕ ℕ).toEmbedding) := by |
| apply Finset.disjoint_left.mpr |
| rintro ⟨i, j⟩ hi hj |
| simp only [YoungDiagram.mem_cells, Saxl.mem_staircase] at hi |
| simp only [Finset.mem_map_equiv, Equiv.prodComm_symm, Equiv.prodComm_apply, |
| Prod.swap_prod_mk, mem_armExtras] at hj |
| omega |
| rw [Finset.card_union_of_disjoint |
| (Finset.disjoint_union_left.mpr ⟨hd, armExtras_disjoint_transpose M b δ hM⟩)] |
| rw [Finset.card_union_of_disjoint (staircase_disjoint_armExtras M b δ (by omega)), |
| Finset.card_map, armExtras_card] |
| simp only [YoungDiagram.card] |
| omega |
|
|
| def candidateLengths (M b δ i : ℕ) : ℕ := |
| if i = 0 then M + b + δ else |
| if i = 1 then M - 1 + b else |
| if i < M - 2 then M - i else |
| if i < M - 1 + b then 2 else |
| if i < M + b + δ then 1 else 0 |
|
|
| lemma candidate_rowLen (M b δ i : ℕ) (hM : 4 ≤ M) : |
| (candidate M b δ).rowLen i = candidateLengths M b δ i := by |
| have hh (j : ℕ) : j < (candidate M b δ).rowLen i ↔ j < candidateLengths M b δ i := by |
| rw [← YoungDiagram.mem_iff_lt_rowLen, mem_candidate] |
| unfold candidateLengths |
| split_ifs < |
| apply le_antisymm |
| · by_contra hn |
| have := (hh _).mp (Nat.lt_of_not_ge hn) |
| omega |
| · by_contra hn |
| have := (hh _).mpr (Nat.lt_of_not_ge hn) |
| omega |
|
|
| lemma candidate_colLen (M b δ i : ℕ) (hM : 4 ≤ M) : |
| (candidate M b δ).colLen i = candidateLengths M b δ i := by |
| rw [← YoungDiagram.rowLen_transpose, candidate_transpose] |
| exact candidate_rowLen M b δ i hM |
|
|
| |
| apply SetLike.ext |
| rintro ⟨i, j⟩ |
| simp only [mem_candidate, Saxl.mem_staircase] |
| omega |
|
|
| def fixedCandidate (n : ℕ) : YoungDiagram := |
| candidate (staircaseIndex n) (remainderPairs n / 2) (remainderPairs n % 2) |
|
|
| |
| (fixedCandidate n).transpose = fixedCandidate n := candidate_transpose _ _ _ |
|
|
| lemma fixedCandidate_eq_staircase_of_remainder_zero (n : ℕ) |
| (hr : remainderPairs n = 0) : |
| fixedCandidate n = Saxl.staircase (staircaseIndex n) := by |
| simp only [fixedCandidate, hr, Nat.zero_div, Nat.zero_mod, candidate_zero] |
|
|
| lemma fixedCandidate_card (n : ℕ) (hM : 3 ≤ staircaseIndex n) : |
| (fixedCandidate n).card = n := by |
| rw [fixedCandidate, candidate_card _ _ _ hM] |
| have hd := degree_eq n |
| omega |
|
|
| theorem fixedCandidate_large_degree (n : ℕ) (hn : 64 < n) : |
| 9 ≤ staircaseIndex n ∧ (fixedCandidate n).card = n ∧ |
| (fixedCandidate n).transpose = fixedCandidate n := by |
| have hM : 9 ≤ staircaseIndex n := by |
| by_contra h |
| exact (not_le_of_gt hn) (degree_le_64_of_small_index n (by omega)) |
| exact ⟨hM, fixedCandidate_card n (by omega), fixedCandidate_transpose n⟩ |
|
|
| end UniversalTensorSquare |
| end |
| end |
|
|
| end OAI |
|
|
