| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Balance.WordPackingGeometry |
| import OAI.RepresentationTheory.UniversalSquare.Band.OutputBands |
|
|
| namespace OAI |
|
|
| /-! Empty Packing. -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace Saxl.Balance |
| open FlagColumns Columns |
|
|
| def WordPacking.empty (d : ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ) : |
| WordPacking [] [] d A label ∅ := by |
| classical |
| let e := Classical.choose (columnShape_placement []) |
| have hr := (Classical.choose_spec (columnShape_placement [])).1 |
| have hc := (Classical.choose_spec (columnShape_placement [])).2 |
| let μ := columnShape [] |
| have h0 : μ.colLen 0 = 0 := by |
| apply Nat.eq_zero_of_not_pos |
| intro h |
| have hmem : (0,0) ∈ μ := YoungDiagram.mem_iff_lt_colLen.mpr h |
| exact isEmptyElim (e.symm ⟨(0,0),hmem⟩) |
| let t := (enumerate []).trans e |
| let f := Fin.castLE (show μ.colLen 0 ≤ d by omega) |
| have he : ∀ a, ((A ∘ pairLetterMap f f) a ∧ |
| (label ∘ pairLetterMap f f) a = 0) ↔ output a ∈ ({0}:Set ℕ) := by |
| intro a |
| have hh := a.isLt |
| simp only [h0,Nat.zero_mul] at hh |
| omega |
| have hs : SupportLE (wordRep 0 0) |
| (projectedSpechtTensor t t (inOutputs {0}) (inOutputs_invariant {0})).toRepresentation := |
| (SupportLE.of_injective (letterLift (Fin.castLE (show 0 ≤ 1 by omega))) |
| (letterLift_injective _ (Fin.castLE_injective _))).trans |
| (constant_output_support t 0 (fun i => Fin.elim0 i)) |
| let w : Fin 0 → Fin 0 := Fin.elim0 |
| let P := boundedPacking t 0 {0} (A ∘ pairLetterMap f f) (label ∘ pairLetterMap f f) |
| he hs (Pi.single w 1) |
| let Q := P.lift f (fun _ => rfl) A label |
| exact { |
| word := w |
| counts := fun j => Fin.elim0 j |
| packing := Q.congr (standard_placed e hr hc (show μ.colLen 0 ≤ d by omega)).symm rfl |
| marksBound := fun i => Fin.elim0 i } |
|
|
| def oneIf (x ε : ℕ) : List ℕ := if ε = 0 then [] else [x] |
|
|
| def markIf (x ε : ℕ) : Finset ℕ := if ε = 0 then ∅ else {x} |
|
|
| theorem WordPacking.optional {x ε d : ℕ} (hx : 0 < x) (hd : x ≤ d) |
| (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ) |
| (he : ε ≠ 0 → ∀ a, (A a ∧ label a = x) ↔ output a = x) : |
| Nonempty (WordPacking (oneIf x ε) (oneIf x ε) d A label (markIf x ε)) := by |
| by_cases hε : ε = 0 |
| · simpa only [oneIf,markIf,ite_eq_left hε] using Nonempty.intro (WordPacking.empty d A label) |
| · obtain ⟨P⟩ := WordPacking.repeated (m := 1) (by decide) hx hd x A label (he hε) |
| simpa only [oneIf,markIf,ite_eq_right hε,List.replicate_one,one_mul] using Nonempty.intro P |
|
|
| end Saxl.Balance |
| end |
| end |
|
|
| end OAI |
|
|
