| import Mathlib |
| import OAI.Probability.BinarySweep.GridBounds.GridChildAverage |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.GridSplit |
| open Signed Irrep Representation |
|
|
| variable {m n h : ℕ} (bits : Fin (m+n) → ℕ) (H : PathFamily bits h) |
| {A : Type*} [Fintype A] [DecidableEq A] |
|
|
| local instance : LinearOrder (GridSlot (rightBits bits)) := rowOrder bits |
| local instance : LinearOrder (GridSlot (leftBits bits)) := columnOrder bits |
| |
| def inputToJunction : FreeSlot H 0 ≃ Fin (junctionSize bits H) := |
| (rowReference bits H).trans (junctionEnum bits H) |
|
|
| def junctionTensorRep (p : A → Bool) : |
| Representation ℂ (Equiv.Perm (FreeSlot H 0)) (TensorSpace A (junctionSize bits H)) := |
| (hilbertTensorRep p (junctionSize bits H)).comp |
| (inputToJunction bits H).permCongrHom.toMonoidHom |
|
|
| def rowChildWeight (z : ℝ) (y : GridSlot (rightBits bits)) |
| (a : Equiv.Perm (Fin (Fintype.card (FreeSlot (rowFamily bits H y) 0)))) : ℂ := |
| conditionalGroupLaw (rowFamily bits H y) z ((Fintype.equivFin _).symm.permCongr a) |
|
|
| def columnChildWeight (z : ℝ) (x : GridSlot (leftBits bits)) |
| (a : Equiv.Perm (Fin (Fintype.card (FreeSlot (columnFamily bits H x) 0)))) : ℂ := |
| conditionalGroupLaw (columnFamily bits H x) z ((Fintype.equivFin _).symm.permCongr a) |
|
|
| def rowMatrix (p : A → Bool) (z : ℝ) := |
| groupedAverage (groupingOrder (rowGrouping bits H)) (groupingPerm (rowGrouping bits H)) p |
| (rowChildWeight bits H z) |
|
|
| def columnMatrix (p : A → Bool) (z : ℝ) := |
| groupedAverage (groupingOrder (columnGrouping bits H)) (groupingPerm (columnGrouping bits H)) p |
| (columnChildWeight bits H z) |
|
|
| lemma row_perm_to_junction (a : ∀y, Equiv.Perm (FreeSlot (rowFamily bits H y) 0)) : |
| (inputToJunction bits H).permCongr |
| ((rowInputEquiv bits H).symm.permCongr (Equiv.Perm.sigmaCongrRight a))= |
| (rowGrouping bits H).permCongr (Equiv.Perm.sigmaCongrRight a) := by |
| ext t |
| simp [inputToJunction,rowGrouping,Equiv.permCongr_apply] |
|
|
| lemma column_perm_to_junction (a : ∀x, Equiv.Perm (FreeSlot (columnFamily bits H x) 0)) : |
| (inputToJunction bits H).permCongr |
| ((rowReference bits H).symm.permCongr |
| ((columnFreeEquiv bits H 0).symm.permCongr (Equiv.Perm.sigmaCongrRight a)))= |
| (columnGrouping bits H).permCongr (Equiv.Perm.sigmaCongrRight a) := by |
| ext t |
| simp [inputToJunction,columnGrouping,Equiv.permCongr_apply] |
|
|
| lemma rowMatrix_linear (p : A → Bool) (z : ℝ) : |
| (rowMatrix bits H p z).toEuclideanLin= |
| ∑a : ∀y, Equiv.Perm (FreeSlot (rowFamily bits H y) 0), |
| (∏y, (conditionalGroupLaw (rowFamily bits H y) z (a y):ℂ)) • |
| hilbertTensorRep p _ ((rowGrouping bits H).permCongr (Equiv.Perm.sigmaCongrRight a)) := by |
| refine Eq.trans (groupedAverage_literal_linear (I := GridSlot (rightBits bits)) |
| (X := fun y => FreeSlot (rowFamily bits H y) 0) (N := junctionSize bits H) |
| (A := A) (rowGrouping bits H) p |
| (fun y a => (conditionalGroupLaw (rowFamily bits H y) z a : ℂ))) ?_ |
| apply Finset.sum_congr |
| · ext a |
| · intro a _ |
|
|
| lemma columnMatrix_linear (p : A → Bool) (z : ℝ) : |
| (columnMatrix bits H p z).toEuclideanLin= |
| ∑a : ∀x, Equiv.Perm (FreeSlot (columnFamily bits H x) 0), |
| (∏x, (conditionalGroupLaw (columnFamily bits H x) z (a x):ℂ)) • |
| hilbertTensorRep p _ ((columnGrouping bits H).permCongr (Equiv.Perm.sigmaCongrRight a)) := by |
| refine Eq.trans (groupedAverage_literal_linear (I := GridSlot (leftBits bits)) |
| (X := fun x => FreeSlot (columnFamily bits H x) 0) (N := junctionSize bits H) |
| (A := A) (columnGrouping bits H) p |
| (fun x a => (conditionalGroupLaw (columnFamily bits H x) z a : ℂ))) ?_ |
| apply Finset.sum_congr |
| · ext a |
| · intro a _ |
|
|
| theorem rowAverage_matrix (p : A → Bool) (z : ℝ) : |
| rowAverage bits H (junctionTensorRep bits H p) z=(rowMatrix bits H p z).toEuclideanLin := by |
| rw [rowAverage_child_laws,rowMatrix_linear] |
| apply Finset.sum_congr |
| · ext a |
| intro a _ |
| congr 1 |
|
|
| theorem columnAverage_matrix (p : A → Bool) (z : ℝ) : |
| columnAverage bits H (junctionTensorRep bits H p) z=(columnMatrix bits H p z).toEuclideanLin := by |
| rw [columnAverage_child_laws,columnMatrix_linear] |
| apply Finset.sum_congr |
| · ext a |
| intro a _ |
| congr 1 |
| change hilbertTensorRep p _ ((inputToJunction bits H).permCongr _) = _ |
| rw [column_perm_to_junction] |
|
|
| end BinaryCoordinateSweeps.GridSplit |
| |
| end |
|
|
| end OAI |
|
|
