| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.StandardTableaux |
|
|
| namespace OAI |
|
|
| section |
| namespace PartialPermutation |
| namespace Tableau |
| noncomputable section |
| open Finset |
|
|
| section Extension |
| variable {α β : Type*} [Fintype α] [PartialOrder α] [Fintype β] [PartialOrder β] |
| |
| def fillingCongr (e : α ≃o β) : StandardFilling α ≃ StandardFilling β where |
| toFun T := ⟨e.symm.toEquiv.trans (T.1.trans (finCongr (Fintype.card_congr e.toEquiv))), |
| (Fin.cast_strictMono (Fintype.card_congr e.toEquiv)).comp (T.2.comp e.symm.strictMono)⟩ |
| invFun T := ⟨e.toEquiv.trans (T.1.trans (finCongr (Fintype.card_congr e.symm.toEquiv))), |
| (Fin.cast_strictMono (Fintype.card_congr e.symm.toEquiv)).comp (T.2.comp e.strictMono)⟩ |
| left_inv T := by apply Subtype.ext; apply Equiv.ext; intro a; apply Fin.ext; simp |
| right_inv T := by apply Subtype.ext; apply Equiv.ext; intro a; apply Fin.ext; simp |
| |
| omit [PartialOrder α] in |
| lemma sum_subtype_card (P : α → Prop) [DecidablePred P] : |
| Fintype.card {a // P a} + Fintype.card {a // ¬P a} = Fintype.card α := by |
| simpa only [Fintype.card_sum] using Fintype.card_congr (Equiv.sumCompl P) |
|
|
| def concatEnumeration (P : α → Prop) [DecidablePred P] |
| (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) : |
| α ≃ Fin (Fintype.card α) := |
| (Equiv.sumCompl P).symm.trans ((Equiv.sumCongr T.1 U.1).trans |
| (finSumFinEquiv.trans (finCongr (sum_subtype_card P)))) |
|
|
| lemma concatEnumeration_pos (P : α → Prop) [DecidablePred P] |
| (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) |
| (a : α) (ha : P a) : (concatEnumeration P T U a).val = (T.1 ⟨a,ha⟩).val := by |
| simp [concatEnumeration, Equiv.sumCompl_symm_apply_of_pos ha] |
|
|
| lemma concatEnumeration_neg (P : α → Prop) [DecidablePred P] |
| (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) |
| (a : α) (ha : ¬P a) : (concatEnumeration P T U a).val = |
| Fintype.card {a // P a} + (U.1 ⟨a,ha⟩).val := by |
| simp [concatEnumeration, Equiv.sumCompl_symm_apply_of_neg ha] |
|
|
| def extendFilling (P : α → Prop) [DecidablePred P] (hP : IsLowerSet {a | P a}) |
| (T : StandardFilling {a // P a}) (U : StandardFilling {a // ¬P a}) : |
| StandardFilling α := by |
| refine ⟨concatEnumeration P T U, ?_⟩ |
| intro a b hab |
| change (concatEnumeration P T U a).val < (concatEnumeration P T U b).val |
| by_cases ha : P a <;> by_cases hb : P b |
| · rw [concatEnumeration_pos P T U a ha, concatEnumeration_pos P T U b hb] |
| exact T.2 (show (⟨a,ha⟩ : {a // P a}) < ⟨b,hb⟩ from hab) |
| · rw [concatEnumeration_pos P T U a ha, concatEnumeration_neg P T U b hb] |
| have h := (T.1 ⟨a,ha⟩).isLt |
| omega |
| · exact (ha (hP hab.le hb)).elim |
| · rw [concatEnumeration_neg P T U a ha, concatEnumeration_neg P T U b hb] |
| exact Nat.add_lt_add_left (U.2 (show (⟨a,ha⟩ : {a // ¬P a}) < ⟨b,hb⟩ from hab)) _ |
|
|
| lemma filling_count_mono_lowerSet (P : α → Prop) [DecidablePred P] |
| (hP : IsLowerSet {a | P a}) : |
| Fintype.card (StandardFilling {a // P a}) ≤ Fintype.card (StandardFilling α) := by |
| let U := chosenFilling {a // ¬P a} |
| apply Fintype.card_le_of_injective (fun T => extendFilling P hP T U) |
| intro T S he |
| apply Subtype.ext |
| apply Equiv.ext |
| intro a |
| apply Fin.ext |
| have h := congrArg (fun T : StandardFilling α => (T.1 a.1).val) he |
| change (concatEnumeration P T U a.1).val = (concatEnumeration P S U a.1).val at h |
| simpa only [concatEnumeration_pos P _ _ _ a.2] using h |
|
|
| lemma filling_count_mono_complement (P : α → Prop) [DecidablePred P] |
| (hP : IsLowerSet {a | P a}) : |
| Fintype.card (StandardFilling {a // ¬P a}) ≤ Fintype.card (StandardFilling α) := by |
| let T := chosenFilling {a // P a} |
| apply Fintype.card_le_of_injective (fun U => extendFilling P hP T U) |
| intro U V he |
| apply Subtype.ext |
| apply Equiv.ext |
| intro a |
| apply Fin.ext |
| have h := congrArg (fun S : StandardFilling α => (S.1 a.1).val) he |
| change (concatEnumeration P T U a.1).val = (concatEnumeration P T V a.1).val at h |
| simp only [concatEnumeration_neg P _ _ _ a.2] at h |
| exact Nat.add_left_cancel h |
|
|
| end Extension |
|
|
| def subdiagramCellsEquiv {μ ν : YoungDiagram} (h : μ ≤ ν) : |
| μ.cells ≃o {x : ν.cells // x.1 ∈ μ.cells} where |
| toFun x := ⟨⟨x.1,h x.2⟩,x.2⟩ |
| invFun x := ⟨x.1.1,x.2⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
| map_rel_iff' := Iff.rfl |
|
|
| lemma standardCount_mono {μ ν : YoungDiagram} (h : μ ≤ ν) : |
| standardCount μ ≤ standardCount ν := by |
| classical |
| have hh : IsLowerSet {x : ν.cells | x.1 ∈ μ.cells} := by |
| intro x y hxy hy |
| exact μ.isLowerSet hxy hy |
| have hcard := Fintype.card_congr (fillingCongr (subdiagramCellsEquiv h)) |
| unfold standardCount |
| rw [hcard] |
| exact filling_count_mono_lowerSet _ hh |
|
|
| end |
| end Tableau |
| end PartialPermutation |
| end |
|
|
| end OAI |
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|
