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e320e82 | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Combinatorics.EditDistance.Families
import OAI.Combinatorics.EditDistance.DirectSums
import OAI.Combinatorics.EditDistance.Slots
namespace OAI
/-!
# Averaged displacement control for recursive word families
-/
namespace EditFamily
open scoped BigOperators
open EditDistortion EditAverage EditFourier EditSlots
section InductiveControl
variable {J : Type} [Fintype J] [DecidableEq J] (i : J)
noncomputable def weight (j : J) : ℝ := (EditSlots.multiplicity i j : ℝ) / (2 * (2 * Fintype.card J - 1))
lemma weight_pos (j : J) : 0 < weight i j := by
have hb : 0 < (Fintype.card J : ℝ) := by
exact_mod_cast Fintype.card_pos_iff.mpr ⟨i⟩
have hb' : (1 : ℝ) ≤ Fintype.card J := by
exact_mod_cast (show 1 ≤ Fintype.card J by exact Fintype.card_pos_iff.mpr ⟨i⟩)
have hc : 0 < (EditSlots.multiplicity i j : ℝ) := by
by_cases hj : j = i <;> simp [EditSlots.multiplicity, hj, hb]
exact div_pos hc (by linarith)
lemma sum_weight : (∑ j : J, weight i j) = 1 / 2 := by
have hb : (1 : ℝ) ≤ Fintype.card J := by
exact_mod_cast (show 1 ≤ Fintype.card J by exact Fintype.card_pos_iff.mpr ⟨i⟩)
simp only [weight, ← Finset.sum_div, sum_multiplicity]
field_simp [show (2 * (Fintype.card J : ℝ) - 1) ≠ 0 by linarith]
variable {p : J → ℕ} {m A : ℕ} {α Q : ℝ}
variable (F : ∀ j, Family (p j) m A α Q)
abbrev ProductState := ∀ j, (F j).State
def lift (j : J) (e : (F j).State) : ProductState F := Pi.single j e
def allPhase : ProductState F := fun j => (F j).v
lemma sum_lift_vectors : (∑ j, lift F j (F j).v) = allPhase F := by
ext j
simp [lift, allPhase]
omit [Fintype J] in
lemma lift_nsmul (j : J) (u : ℕ) (e : (F j).State) :
lift F j (u • e) = u • lift F j e := by
ext k
by_cases hk : k = j
· subst k; simp [lift]
· simp [lift, hk]
lemma variation_zero (H : ProductState F → L1) : variation H 0 = 0 := by
simp [variation, avg]
lemma component_control (hQ : 0 ≤ Q) (H : ProductState F → L1)
(hh : ∀ j e r, (e,r) ∈ (F j).steps →
variation H (lift F j e) ≤ weight i j * r)
(j : J) (a : ℕ) (ha : a < p j) :
variation H (a • lift F j (F j).v) ≤ weight i j * Q := by
by_cases ha0 : a = 0
· simp only [ha0, zero_nsmul, variation_zero]
exact mul_nonneg (weight_pos i j).le hQ
let T := EditSlices.slice (fun j => (F j).State) H j (weight i j)
have hT : ∀ e r, (e,r) ∈ (F j).steps → variation T e ≤ r := by
intro e r her
rw [EditSlices.variation_slice _ H j _ (weight_pos i j)]
apply (div_le_iff₀ (weight_pos i j)).mpr
simpa only [lift, mul_comm] using hh j e r her
have hC := (F j).control T hT a (Nat.pos_of_ne_zero ha0) ha
rw [EditSlices.variation_slice _ H j _ (weight_pos i j)] at hC
have hC' := (div_le_iff₀ (weight_pos i j)).mp hC
change variation H (lift F j (a • (F j).v)) ≤ Q * weight i j at hC'
rw [lift_nsmul] at hC'
simpa only [mul_comm] using hC'
theorem successor_control
(hp : ∀ j, (p j).Prime) (hinj : Function.Injective p)
(P : ℝ) (hP : 1 ≤ 2 * P) (hpP : ∀ j, (p j : ℝ) ≤ 2 * P)
(h : ℕ) (hh : 0 < h) (L : ℕ) (hQ : 0 ≤ Q)
(H : ProductState F → L1) (hτ : variation H (allPhase F) ≤ 2 / L)
(hlift : ∀ j e r, (e,r) ∈ (F j).steps →
variation H (lift F j e) ≤ weight i j * r)
(u : ℕ) (hu : u < p i) :
variation H (u • lift F i (F i).v) ≤
2 * (2 * P) ^ (2 * h) / L + Q / h := by
let (j : J) : NeZero (p j) := ⟨(hp j).ne_zero⟩
have hv (j : J) : p j • lift F j (F j).v = 0 := by
rw [← lift_nsmul, (F j).annihilate]
simp [lift]
have hc := prime_vector_displacement p (fun j => lift F j (F j).v) hv hp hinj
P hP hpP h hh H i (u : ZMod (p i))
rw [ZMod.val_natCast_of_lt hu, sum_lift_vectors] at hc
have hb (j : J) : avg (fun a : ZMod (p j) => variation H (a.val • lift F j (F j).v)) ≤
weight i j * Q := by
rw [← avg_const (X := ZMod (p j)) (weight i j * Q)]
exact avg_mono (fun a => component_control i F hQ H hlift j a.val (ZMod.val_lt a))
calc
_ ≤ (2 * P) ^ (2 * h) * variation H (allPhase F) +
(2 / (h : ℝ)) * ∑ j, avg (fun a : ZMod (p j) =>
variation H (a.val • lift F j (F j).v)) := hc
_ ≤ (2 * P) ^ (2 * h) * (2 / (L : ℝ)) +
(2 / (h : ℝ)) * ∑ j, weight i j * Q := by
apply add_le_add
· exact mul_le_mul_of_nonneg_left hτ (pow_nonneg (by linarith) _)
· exact mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun j _ => hb j)) (by positivity)
_ = _ := by rw [← Finset.sum_mul, sum_weight]; ring
end InductiveControl
end EditFamily
end OAI
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