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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Specht.ZeroColumns
import OAI.RepresentationTheory.UniversalSquare.Balance.BalancePacking
namespace OAI
/-! Word Contents. -/
section
noncomputable section
namespace Saxl
open scoped BigOperators
def wordContent {n d : ℕ} (w : Fin n → Fin d) (j : Fin d) : ℕ :=
(Finset.univ.filter (fun i => w i = j)).card
lemma wordContent_position_join {n a b d e E : ℕ}
(p : Fin n ≃ Fin a ⊕ Fin b) (q : Fin E ≃ Fin d ⊕ Fin e)
(w : Fin a → Fin d) (z : Fin b → Fin e) (j : Fin E) :
wordContent
((positionWordsEquiv p).symm ((fun i => q.symm (Sum.inl (w i))),
(fun i => q.symm (Sum.inr (z i))))) j =
Sum.elim (wordContent w) (wordContent z) (q j) := by
classical
rw [wordContent, Finset.card_filter]
rw [← Equiv.sum_comp p.symm]
rw [Fintype.sum_sum_type]
rcases hj : q j with k | k
· have hqq : j = q.symm (Sum.inl k) := (q.eq_symm_apply).mpr hj
subst j
simp only [positionWordsEquiv, Equiv.coe_fn_symm_mk, joinPositions, Equiv.apply_symm_apply, Sum.elim_inl,
Sum.elim_inr, q.symm.injective.eq_iff,
Sum.inl.injEq, Sum.inr_ne_inl, ite_false, Finset.sum_const_zero, add_zero]
rw [wordContent,Finset.card_filter]
· have hqq : j = q.symm (Sum.inr k) := (q.eq_symm_apply).mpr hj
subst j
simp only [positionWordsEquiv, Equiv.coe_fn_symm_mk, joinPositions, Equiv.apply_symm_apply, Sum.elim_inl,
Sum.elim_inr, q.symm.injective.eq_iff,
Sum.inr.injEq, Sum.inl_ne_inr, ite_false, Finset.sum_const_zero, zero_add]
rw [wordContent,Finset.card_filter]
lemma wordContent_lift_equiv {n d e : ℕ} (q : Fin d ≃ Fin e)
(w : Fin n → Fin d) (j : Fin e) :
wordContent (q ∘ w) j = wordContent w (q.symm j) := by
classical
unfold wordContent
congr 1
apply Finset.filter_congr
intro i _
change q (w i) = j ↔ w i = q.symm j
exact q.eq_symm_apply.symm
def appendLetters (ps qs : List ℕ) : Fin (ps ++ qs).length ≃ Fin ps.length ⊕ Fin qs.length :=
(finCongr (List.length_append (as := ps) (bs := qs))).trans finSumFinEquiv.symm
lemma get_appendLetters (ps qs : List ℕ) (j : Fin (ps ++ qs).length) :
(ps ++ qs).get j = Sum.elim ps.get qs.get (appendLetters ps qs j) := by
obtain ⟨j,rfl⟩ := (appendLetters ps qs).symm.surjective j
rcases j with j | j
· simp only [Equiv.apply_symm_apply, Sum.elim_inl]
change (ps ++ qs)[j.val] = ps[j.val]
exact List.getElem_append_left j.isLt
· simp only [Equiv.apply_symm_apply, Sum.elim_inr]
change (ps ++ qs)[ps.length+j.val] = qs[j.val]
rw [List.getElem_append_right (by omega)]
simp
lemma list_get_fiber_card (ps : List ℕ) (k : ℕ) :
Fintype.card {i : Fin ps.length // ps.get i = k} = ps.count k := by
classical
rw [Fintype.card_subtype, Finset.card_filter]
induction ps with
| nil => simp
| cons p ps ih =>
change (∑ i : Fin (ps.length+1), if (p::ps).get i = k then (1:ℕ) else 0) = _
rw [Fin.sum_univ_succ]
change (if p = k then 1 else 0) + (∑ i : Fin ps.length, if ps.get i = k then 1 else 0) = _
rw [ih]
simp [List.count_cons, Nat.add_comm]
theorem list_get_equiv {ps qs : List ℕ} (h : ps.Perm qs) :
∃ q : Fin ps.length ≃ Fin qs.length, ∀ i, qs.get (q i) = ps.get i := by
classical
let f (k : ℕ) : {i : Fin ps.length // ps.get i = k} ≃
{i : Fin qs.length // qs.get i = k} :=
Fintype.equivOfCardEq (by rw [list_get_fiber_card,list_get_fiber_card,h.count_eq])
let q := (Equiv.sigmaFiberEquiv ps.get).symm.trans
((Equiv.sigmaCongrRight f).trans (Equiv.sigmaFiberEquiv qs.get))
exact ⟨q,fun i => (f (ps.get i) ⟨i,rfl⟩).property⟩
lemma columnShape_rows_permutation (ps : List ℕ) (hp : ∀ p ∈ ps, 0 < p) :
ps.Perm (Columns.columnShape ps).transpose.rowLens := by
rw [Columns.columnShape_cols]
have he : ps.filter (fun p => 0 < p) = ps := by
apply List.filter_eq_self.mpr
intro p h
simpa using hp p h
simpa only [he] using Columns.positiveSorted_perm ps
end Saxl
end
end
end OAI
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