| 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 | |