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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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import Mathlib
import OAI.Probability.BinarySweep.Conditional.AugmentedBlock
import OAI.Probability.BinarySweep.MatrixBounds.HilbertBlockTrace
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
open Irrep Representation
variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h)
{A W : Type*} [Fintype A] [NormedAddCommGroup W] [InnerProductSpace ℂ W]
[FiniteDimensional ℂ W] (e : A ⊕ Fin k ≃ FreeSlot H 0)
(σ : Representation ℂ (Equiv.Perm A) W)
abbrev inducedPlacementAverage (z : ℝ) :=
groupAverage (coindHilbertRep (blockPerm e) σ
(fun x => (placementSection e x)⁻¹) (placement_cosets_bijective e))
(fun a => (conditionalGroupLaw H z a:ℂ))
def placementFiberMoment (z : ℝ) (q : ℕ) (x y : Placement H k 0) : ℝ :=
if hp : 0<placementKernel H z x y then
let g := realizePlacement H z x (placementIdentification H k y) hp
evenMoment q (groupAverage (augmentedRepresentation H x g e σ)
(fun a => (conditionalGroupLaw (augmentByChoice H x g) z a:ℂ)))
else 0
lemma realizedOutputIndex (z : ℝ) (x y : Placement H k 0)
(hp : 0<placementKernel H z x y) :
augmentedOutputIndex H x (realizePlacement H z x (placementIdentification H k y) hp)=y := by
apply (placementIdentification H k).injective
rw [←placementBijection_index]
exact realizePlacement_spec H z x (placementIdentification H k y) hp
lemma inducedPlacementBlock_root {z : ℝ} (hz : 0≤z) (hz1 : z<1)
(hσ : ∀ g v, ‖σ g v‖=‖v‖) {q : ℕ} (hq : 0<q)
(x y : Placement H k 0) :
evenMoment q (hilbertBlock (inducedPlacementAverage H e σ z) y x)^(((2*q:ℕ):ℝ)⁻¹) =
placementKernel H z x y * (placementFiberMoment H e σ z q x y)^(((2*q:ℕ):ℝ)⁻¹) := by
by_cases hp : 0<placementKernel H z x y
· unfold placementFiberMoment
rw [dite_eq_left hp]
have he := placementBlock_moment_root H e σ hz hz1 hσ hq x
(realizePlacement H z x (placementIdentification H k y) hp)
rw [realizedOutputIndex H z x y hp] at he
exact he
· have he : placementKernel H z x y=0 :=
le_antisymm (le_of_not_gt hp) (placementKernel_nonneg H hz hz1 x y)
rw [he,zero_mul,placementBlock_zero H e σ hz hz1 x y he]
have hq0 : q≠0 := hq.ne'
simp [evenMoment,hq0]
theorem induced_placement_cycle_bound {z : ℝ} (hz : 0≤z) (hz1 : z<1)
(hσ : ∀ g v, ‖σ g v‖=‖v‖) {q : ℕ} (hq : 0<q) :
evenMoment q (inducedPlacementAverage H e σ z) ≤
∑x : Fin (2*q) → Placement H k 0,
(∏j, placementEdge H z (decide (j.val%2≠0)) (x j) (x (finRotate (2*q) j))) *
∏j, (if j.val%2=0 then
placementFiberMoment H e σ z q (x j) (x (finRotate (2*q) j))
else placementFiberMoment H e σ z q (x (finRotate (2*q) j)) (x j))^(((2*q:ℕ):ℝ)⁻¹) := by
refine (evenMoment_blocks_le hq (inducedPlacementAverage H e σ z)).trans_eq ?_
apply Finset.sum_congr rfl
intro x _
rw [←Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro j _
by_cases he : j.val%2=0
· simp only [placementEdge,decide_eq_true_eq,ne_eq,he,not_true_eq_false,ite_false,ite_true]
exact inducedPlacementBlock_root H e σ hz hz1 hσ hq _ _
· simp only [placementEdge,decide_eq_true_eq,ne_eq,he,not_false_eq_true,ite_true,ite_false]
exact inducedPlacementBlock_root H e σ hz hz1 hσ hq _ _
end BinaryCoordinateSweeps
end
end OAI
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