| 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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|
