File size: 3,347 Bytes
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 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 | 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
|