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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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