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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.GridBounds.GroupedTypes
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.Signed
variable {I : Type*} [Fintype I] [LinearOrder I]
{X : I → Type*} [∀i,Fintype (X i)] {N : ℕ}
omit [LinearOrder I] in
theorem groupingCard (e : (Σi, X i) ≃ Fin N) :
Fintype.card (Σₗ i, Fin (Fintype.card (X i)))=N := by
change Fintype.card (Σi, Fin (Fintype.card (X i)))=N
have he := Fintype.card_congr e
simpa only [Fintype.card_sigma,Fintype.card_fin] using he
def groupingOrder (e : (Σi, X i) ≃ Fin N) :
(Σₗ i, Fin (Fintype.card (X i))) ≃o Fin N :=
(Fintype.orderIsoFinOfCardEq _ (groupingCard e)).symm
def groupingPerm (e : (Σi, X i) ≃ Fin N) : Equiv.Perm (Fin N) :=
((groupingOrder e).symm.toEquiv.trans ofLex).trans
((Equiv.sigmaCongrRight (fun i => (Fintype.equivFin (X i)).symm)).trans e)
lemma groupingPerm_apply (e : (Σi, X i) ≃ Fin N) (i : I) (a : Fin (Fintype.card (X i))) :
groupingPerm e (groupingOrder e (toLex ⟨i,a⟩))=e ⟨i,(Fintype.equivFin (X i)).symm a⟩ := by
simp [groupingPerm,Equiv.sigmaCongrRight]
@[simp] lemma groupingPerm_inv_apply (e : (Σi, X i) ≃ Fin N) (i : I) (x : X i) :
(groupingPerm e)⁻¹ (e ⟨i,x⟩)=groupingOrder e (toLex ⟨i,Fintype.equivFin (X i) x⟩) := by
apply (groupingPerm e).injective
rw [groupingPerm_apply,Equiv.symm_apply_apply]
exact (groupingPerm e).apply_symm_apply _
lemma grouping_groupOf (e : (Σi, X i) ≃ Fin N) (t : Fin N) :
groupOf (groupingOrder e) (groupingPerm e) t=(e.symm t).1 := by
simp [groupOf,groupingPerm,Equiv.sigmaCongrRight]
lemma grouping_permutation (e : (Σi, X i) ≃ Fin N)
(a : ∀i, Equiv.Perm (X i)) :
e.permCongr (Equiv.Perm.sigmaCongrRight a)=
groupingPerm e*blockPerm (groupingOrder e)
(fun i => (Fintype.equivFin (X i)).permCongr (a i))*(groupingPerm e)⁻¹ := by
apply Equiv.ext
intro t
obtain ⟨⟨i,x⟩,rfl⟩ := e.surjective t
simp only [Equiv.Perm.mul_apply,groupingPerm_inv_apply,blockPerm_apply,groupingPerm_apply,
Equiv.permCongr_apply,Equiv.symm_apply_apply]
rfl
end BinaryCoordinateSweeps.Signed
end
end OAI
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