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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
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