openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Balance /WordPackingGeometry.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Balance.ConstantOutputs | |
| import OAI.RepresentationTheory.UniversalSquare.Balance.WordPackingLeaves | |
| namespace OAI | |
| /-! Word Packing Geometry. -/ | |
| section | |
| noncomputable section | |
| namespace Saxl.Balance | |
| open FlagColumns Columns | |
| theorem WordPacking.of_bounded_shape {rs ps : List ℕ} {μ : YoungDiagram} {d E : ℕ} | |
| (hp : rs.Perm μ.transpose.rowLens) (hd : μ.colLen 0 ≤ d) | |
| (hsum : ps.sum = rs.sum) (hlen : ps.length ≤ E) | |
| (k : ℕ) (I : Set ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ) | |
| (he : ∀ a, (A a ∧ label a = k) ↔ output a ∈ I) | |
| (hs : ∀ t : Tableau rs.sum μ, SupportLE (wordRep rs.sum E) | |
| (projectedSpechtTensor t t (inOutputs I) (inOutputs_invariant I)).toRepresentation) : | |
| Nonempty (WordPacking rs ps d A label {k}) := by | |
| obtain ⟨e,hr,hc⟩ := place_columns μ hp | |
| exact ⟨WordPacking.bounded e hr hc hd hsum hlen k I A label he (hs _)⟩ | |
| theorem WordPacking.neighbor {r d : ℕ} (hr : 0 < r) (hd : r+1 ≤ d) | |
| {ps : List ℕ} (hsum : ps.sum = 2*r+1) (hlen : ps.length ≤ 4) | |
| (k : ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ) | |
| (he : ∀ a, (A a ∧ label a = k) ↔ output a ∈ Set.Icc r (r+1)) : | |
| Nonempty (WordPacking [r+1,r] ps d A label {k}) := by | |
| have hrow : (ShortColumns.shape r 1).rowLens = [r+1,r] := | |
| YoungDiagram.rowLens_ofRowLens_eq_self (by simp; omega) | |
| have hD : (ShortColumns.shape r 1).transpose.colLen 0 ≤ d := by | |
| rw [YoungDiagram.colLen_transpose] | |
| have hh := YoungDiagram.rowLen_ofRowLens (hw := by apply List.Pairwise.sortedGE; simp) | |
| (show Fin [r+1,r].length from ⟨0,by simp⟩) | |
| change (ShortColumns.shape r 1).rowLen 0 = r+1 at hh | |
| rw [hh] | |
| exact hd | |
| apply WordPacking.of_bounded_shape (μ := (ShortColumns.shape r 1).transpose) | |
| (by rw [YoungDiagram.transpose_transpose,hrow]) hD (by simp; omega) hlen k _ A label he | |
| intro t | |
| exact neighboring_word_support (by simp; omega) t | |
| lemma replicate_sorted (m k : ℕ) : (List.replicate m k).SortedGE := | |
| List.Pairwise.sortedGE (List.pairwise_replicate.mpr (Or.inr le_rfl)) | |
| lemma repeated_column_length {m k n : ℕ} (hk : 0 < k) | |
| (t : Tableau n (YoungDiagram.ofRowLens (List.replicate m k) (replicate_sorted m k)).transpose) | |
| (i : Fin n) : (columnLengthWord t i).val+1 = k := by | |
| have hm := YoungDiagram.mem_transpose.mp (t i).property | |
| obtain ⟨hj,hi⟩ := YoungDiagram.mem_ofRowLens.mp hm | |
| have he := YoungDiagram.rowLen_ofRowLens (hw := replicate_sorted m k) | |
| (show Fin (List.replicate m k).length from ⟨(t i).val.2,hj⟩) | |
| simp only [Fin.getElem_fin, List.getElem_replicate] at he | |
| change (YoungDiagram.ofRowLens (List.replicate m k) (replicate_sorted m k)).transpose.colLen | |
| (t i).val.2 - 1 + 1 = k | |
| rw [YoungDiagram.colLen_transpose,he] | |
| omega | |
| theorem WordPacking.repeated {m k d : ℕ} (hm : 0 < m) (hk : 0 < k) (hd : k ≤ d) | |
| (j : ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ) | |
| (he : ∀ a, (A a ∧ label a = j) ↔ output a = k) : | |
| Nonempty (WordPacking (List.replicate m k) [m*k] d A label {j}) := by | |
| let μ := (YoungDiagram.ofRowLens (List.replicate m k) (replicate_sorted m k)).transpose | |
| have hrow : μ.transpose.rowLens = List.replicate m k := by | |
| rw [YoungDiagram.transpose_transpose] | |
| apply YoungDiagram.rowLens_ofRowLens_eq_self | |
| intro x hx | |
| rw [(List.mem_replicate.mp hx).2] | |
| exact hk | |
| have hD : μ.colLen 0 ≤ d := by | |
| rw [YoungDiagram.colLen_transpose] | |
| have hh := YoungDiagram.rowLen_ofRowLens (hw := replicate_sorted m k) | |
| (show Fin (List.replicate m k).length from ⟨0,by simpa⟩) | |
| simp only [Fin.getElem_fin, List.getElem_replicate] at hh | |
| rw [hh] | |
| exact hd | |
| apply WordPacking.of_bounded_shape (μ := μ) (E := 1) (by rw [hrow]) hD | |
| (by simp) (by simp) j {k} A label he | |
| intro t | |
| exact constant_output_support t k (repeated_column_length hk t) | |
| end Saxl.Balance | |
| end | |
| end | |
| end OAI | |