| import Mathlib |
| import OAI.Probability.BinarySweep.Conditional.PlacementCycleMoment |
| import OAI.Probability.BinarySweep.Conditional.WeightedCycle |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep Representation |
|
|
| variable {b h k r : 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) |
|
|
| lemma placement_weighted_cycle_bound_pos (hd : ∀j, bits j≤2*r) {z : ℝ} |
| (hz : 0≤z) (hzr : z≤linePerturbationRadius r) |
| {N : ℕ} (hN : 0<N) (reverse : Fin N → Bool) : |
| (∑x : Fin N → Placement H k 0, ∏i, |
| placementEdge H z (reverse i) (x i) (x (finRotate N i))* |
| Real.exp (-placementEdgeCost H z (reverse i) (x i) (x (finRotate N i))/(N:ℝ))) ≤ |
| Real.exp (-pathCost H+Real.log 4*k*b) := by |
| obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hN.ne' |
| simpa only [Nat.cast_add,Nat.cast_one,Nat.succ_eq_add_one] using |
| placement_weighted_cycle_bound H hd hz hzr k n reverse |
|
|
| omit [Fintype A] in |
| lemma placementFiberMoment_nonneg (z : ℝ) (q : ℕ) (x y : Placement H k 0) : |
| 0≤placementFiberMoment H e σ z q x y := by |
| unfold placementFiberMoment |
| split_ifs |
| · exact evenMoment_nonneg _ _ |
| · rfl |
|
|
| theorem induced_placement_of_hook_bound (hd : ∀j, bits j≤2*r) {z : ℝ} |
| (hz : 0≤z) (hzr : z≤linePerturbationRadius r) |
| (hσ : ∀g v, ‖σ g v‖=‖v‖) {q : ℕ} (hq : 0<q) (B : ℝ) |
| (hf : ∀x y : Placement H k 0, 0<placementKernel H z x y → |
| placementFiberMoment H e σ z q x y ≤ |
| Real.exp (B-placementTransitionCost H z x (placementIdentification H k y))) : |
| evenMoment q (inducedPlacementAverage H e σ z) ≤ |
| Real.exp (B-pathCost H+Real.log 4*k*b) := by |
| have hz1 := hzr.trans_lt (linePerturbationRadius_lt_one r) |
| have hqR : (0:ℝ)<(2*q:ℕ) := by positivity |
| have hf' (x y : Placement H k 0) : |
| (placementFiberMoment H e σ z q x y)^(((2*q:ℕ):ℝ)⁻¹) ≤ |
| Real.exp (B/((2*q:ℕ):ℝ))* |
| Real.exp (-placementTransitionCost H z x (placementIdentification H k y)/((2*q:ℕ):ℝ)) := by |
| have hle : placementFiberMoment H e σ z q x y ≤ |
| Real.exp (B-placementTransitionCost H z x (placementIdentification H k y)) := by |
| by_cases hp : 0<placementKernel H z x y |
| · exact hf x y hp |
| · simp only [placementFiberMoment,dite_eq_right hp] |
| exact (Real.exp_pos _).le |
| refine (Real.rpow_le_rpow (placementFiberMoment_nonneg H e σ z q x y) hle |
| (inv_nonneg.mpr hqR.le)).trans_eq ?_ |
| rw [←Real.exp_mul,←Real.exp_add] |
| congr 1 |
| ring |
| have hconst : (∏_j : Fin (2*q), Real.exp (B/((2*q:ℕ):ℝ)))=Real.exp B := by |
| rw [Finset.prod_const,Finset.card_univ,Fintype.card_fin,←Real.exp_nat_mul] |
| congr 1 |
| field_simp |
| let rev (j : Fin (2*q)) := decide (j.val%2≠0) |
| have hedge (x : Fin (2*q) → Placement H k 0) (j : Fin (2*q)) : |
| (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) ≤ |
| Real.exp (B/((2*q:ℕ):ℝ))* |
| Real.exp (-placementEdgeCost H z (rev j) (x j) (x (finRotate (2*q) j))/((2*q:ℕ):ℝ)) := by |
| by_cases hj : j.val%2=0 |
| · simpa only [rev,placementEdgeCost,decide_eq_true_eq,ne_eq,hj,not_false_eq_true,not_true_eq_false, |
| ite_true,ite_false] using hf' (x j) (x (finRotate (2*q) j)) |
| · simpa only [rev,placementEdgeCost,decide_eq_true_eq,ne_eq,hj,not_false_eq_true,not_true_eq_false, |
| ite_true,ite_false] using hf' (x (finRotate (2*q) j)) (x j) |
| have hs := induced_placement_cycle_bound H e σ hz hz1 hσ hq |
| refine hs.trans ?_ |
| calc |
| _ ≤ ∑x : Fin (2*q) → Placement H k 0, |
| (∏j, placementEdge H z (rev j) (x j) (x (finRotate (2*q) j)))* |
| ∏j, (Real.exp (B/((2*q:ℕ):ℝ))* |
| Real.exp (-placementEdgeCost H z (rev j) (x j) (x (finRotate (2*q) j))/((2*q:ℕ):ℝ))) := by |
| apply Finset.sum_le_sum |
| intro x _ |
| apply mul_le_mul_of_nonneg_left |
| · apply Finset.prod_le_prod₀ |
| · intro j _ |
| apply Real.rpow_nonneg |
| split_ifs <;> exact placementFiberMoment_nonneg H e σ z q _ _ |
| · intro j _; exact hedge x j |
| · apply Finset.prod_nonneg |
| intro j _ |
| unfold placementEdge |
| split_ifs <;> exact placementKernel_nonneg H hz hz1 _ _ |
| _ = Real.exp B * (∑x : Fin (2*q) → Placement H k 0, ∏j, |
| placementEdge H z (rev j) (x j) (x (finRotate (2*q) j))* |
| Real.exp (-placementEdgeCost H z (rev j) (x j) (x (finRotate (2*q) j))/((2*q:ℕ):ℝ))) := by |
| simp_rw [Finset.prod_mul_distrib,hconst] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
| _ ≤ Real.exp B * Real.exp (-pathCost H+Real.log 4*k*b) := |
| mul_le_mul_of_nonneg_left (placement_weighted_cycle_bound_pos H hd hz hzr |
| (by omega) rev) (Real.exp_pos B).le |
| _ = _ := by rw [←Real.exp_add]; congr 1; ring |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
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| end OAI |
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