| import Mathlib |
| import OAI.Probability.BinarySweep.Conditional.ConditionalAverage |
| import OAI.Probability.BinarySweep.MatrixBounds.CenteredMatrix |
| import OAI.Probability.BinarySweep.SparseBounds.SparseKernel |
| import OAI.Probability.BinarySweep.Trajectories.EndpointPlacement |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Young Sparse |
|
|
| variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
|
|
| lemma conditionalGroupLaw_expectation {W : Type*} [AddCommMonoid W] [Module ℝ W] |
| (z : ℝ) (f : Equiv.Perm (FreeSlot H 0) → W) : |
| (∑a, conditionalGroupLaw H z a • f a) = |
| ∑g : ConditionalChoices H, conditionalChoiceWeight H z g • f (remainingPerm H g.val g.property) := by |
| unfold conditionalGroupLaw |
| simp only [Finset.sum_smul] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro g _ |
| simp only [ite_smul,zero_smul] |
| simp |
|
|
| lemma partialKernel_event (g : ConditionalChoices H) (A : Finset (Fin k)) |
| (x y : Placement H k 0) : |
| subPlacement A (movePlacement (remainingPerm H g.val g.property) x)=subPlacement A y ↔ |
| ∀i∈A, gridSweep bits g.val (x i).val=((placementIdentification H k y i).val) := by |
| constructor |
| · intro he i hi |
| have hh := congrArg (fun u => u ⟨i,hi⟩) he |
| have he' : remainingPerm H g.val g.property (x i)=y i := hh |
| have ht := congrArg (freeIdentification H) he' |
| change (freeIdentification H) ((freeIdentification H).symm |
| (remainingBijection H g.val g.property (x i)))=(freeIdentification H) (y i) at ht |
| rw [Equiv.apply_symm_apply] at ht |
| exact congrArg Subtype.val ht |
| · intro he |
| apply Function.Embedding.ext |
| intro i |
| change (freeIdentification H).symm (remainingBijection H g.val g.property (x i.val))=y i.val |
| apply (freeIdentification H).injective |
| rw [Equiv.apply_symm_apply] |
| apply Subtype.ext |
| exact he i.val i.property |
|
|
| lemma partialKernel_conditional (z : ℝ) (A : Finset (Fin k)) (x y : Placement H k 0) : |
| (∑a : Equiv.Perm (FreeSlot H 0), (conditionalGroupLaw H z a:ℂ) * partialKernel A a y x)= |
| (endpointProbability H z A (fun i => ((x i).val,(placementIdentification H k y i).val)):ℂ) := by |
| have he := conditionalGroupLaw_expectation H z (fun a => partialKernel A a y x) |
| simp only [Complex.real_smul] at he |
| rw [he,endpointProbability_conditional] |
| unfold conditionalEventWeight |
| rw [Complex.ofReal_sum] |
| apply Finset.sum_congr |
| · ext g; simp only [Finset.mem_univ] |
| intro g _ |
| simp only [partialKernel,partialKernel_event H g A x y] |
| split_ifs <;> simp |
| |
| theorem centeredMatrix_conditional (z : ℝ) (x y : Placement H k 0) : |
| centeredMatrix (gridSize bits) (fun a => (conditionalGroupLaw H z a:ℂ)) y x = |
| (centeredEndpointKernel H z (fun i => ((x i).val,(placementIdentification H k y i).val)):ℂ) := by |
| simp only [centeredMatrix,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul] |
| simp_rw [partialKernel_conditional H z] |
| unfold centeredEndpointKernel alternatingSubsetSum |
| simp only [Complex.ofReal_mul,Complex.ofReal_pow,Complex.ofReal_inv,Complex.ofReal_natCast, |
| Complex.ofReal_sum,Complex.ofReal_neg,Complex.ofReal_one,Finset.card_univ] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro A _ |
| simp only [div_eq_mul_inv,inv_pow] |
| ring |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| end OAI |
|
|
