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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import OAI.Combinatorics.EditDistance.FourierAverages
import OAI.Combinatorics.EditDistance.Cuts
namespace OAI
/-!
# Prime-product displacement for absolutely summable maps
-/
namespace EditFourier
open scoped BigOperators
open Finset EditAverage
noncomputable def variation {G E : Type*} [AddCommGroup G] [Fintype G]
[NormedAddCommGroup E] (H : G → E) (l : G) : ℝ :=
avg (fun z => ‖H (z + l) - H z‖)
section CutTransport
variable {G : Type*} [AddCommGroup G] [Fintype G] [DecidableEq G]
lemma quadraticVariation_cut (A : Finset G) (l : G) :
quadraticVariation (fun z => (EditCuts.cut A z : ℂ)) l =
variation (EditCuts.cut A) l := by
unfold quadraticVariation variation avg
congr 1
apply sum_congr rfl
intro z _
by_cases hz : z ∈ A <;> by_cases hzl : z + l ∈ A <;>
simp [EditCuts.cut, hz, hzl]
lemma variation_cut_decomposition (H : G → EditCuts.RealL1) (c : Finset G → ℝ)
(hd : ∀ x y, ‖H x - H y‖ = ∑ A : Finset G, c A * |EditCuts.cut A x - EditCuts.cut A y|)
(l : G) : variation H l = ∑ A : Finset G, c A * variation (EditCuts.cut A) l := by
simp_rw [variation, hd, avg_sum, avg_mul, Real.norm_eq_abs]
end CutTransport
section PrimeProductL1
variable {J : Type*} [Fintype J] [DecidableEq J]
variable (p : J → ℕ) [∀ j, NeZero (p j)]
variable {G : Type*} [AddCommGroup G] [Fintype G]
theorem prime_product_displacement
(hcop : Pairwise (fun i j => (p i).Coprime (p j)))
(B : ℝ) (hB : 1 ≤ B) (hpB : ∀ j, (p j : ℝ) ≤ B)
(h : ℕ) (hh : 0 < h) (φ : (∀ j, ZMod (p j)) →+ G)
(H : G → EditCuts.RealL1) (i : J) (u : ZMod (p i)) :
variation H (φ (Pi.single i u)) ≤
B ^ (2 * h) * variation H (φ 1) + (2 / (h : ℝ)) *
∑ j, avg (fun a : ZMod (p j) => variation H (φ (Pi.single j a))) := by
classical
have hcut (A : Finset G) :
variation (EditCuts.cut A) (φ (Pi.single i u)) ≤
B ^ (2 * h) * variation (EditCuts.cut A) (φ 1) + (2 / (h : ℝ)) *
∑ j, avg (fun a : ZMod (p j) => variation (EditCuts.cut A) (φ (Pi.single j a))) := by
simpa only [quadraticVariation_cut] using group_quadratic_comparison p hcop B hB hpB h hh φ
(fun z => (EditCuts.cut A z : ℂ)) i u
obtain ⟨c, hc, _, _, hd⟩ := EditCuts.l1_decomposition H
simp_rw [variation_cut_decomposition H c hd]
calc
_ ≤ ∑ A : Finset G, c A *
(B ^ (2 * h) * variation (EditCuts.cut A) (φ 1) + (2 / (h : ℝ)) *
∑ j, avg (fun a : ZMod (p j) => variation (EditCuts.cut A) (φ (Pi.single j a)))) := by
apply sum_le_sum
intro A _
exact mul_le_mul_of_nonneg_left (hcut A) (hc A)
_ = _ := by
simp_rw [mul_add, sum_add_distrib, avg_sum, avg_mul]
congr 1
· simp_rw [mul_left_comm _ (B ^ (2 * h)), ← mul_sum]
· simp_rw [mul_left_comm _ (2 / (h : ℝ)), ← mul_sum, mul_sum]
rw [sum_comm]
end PrimeProductL1
end EditFourier
namespace EditFourier
open scoped BigOperators
open Finset EditAverage
def torsionHom {G : Type*} [AddCommGroup G] (p : ℕ) (v : G) (hv : p • v = 0) :
ZMod p →+ G :=
ZMod.lift p ⟨{ toFun := fun z : ℤ => z • v
map_zero' := zero_zsmul v
map_add' := fun x y => add_zsmul v x y }, by simpa using hv⟩
lemma torsionHom_natCast {G : Type*} [AddCommGroup G] (p : ℕ) (v : G) (hv : p • v = 0)
(a : ℕ) : torsionHom p v hv (a : ZMod p) = a • v := by
simpa [torsionHom] using ZMod.lift_coe p
(⟨{ toFun := fun z : ℤ => z • v
map_zero' := zero_zsmul v
map_add' := fun x y => add_zsmul v x y }, by simpa using hv⟩) (a : ℤ)
lemma torsionHom_apply {G : Type*} [AddCommGroup G] (p : ℕ) [NeZero p]
(v : G) (hv : p • v = 0) (a : ZMod p) : torsionHom p v hv a = a.val • v := by
rw [← torsionHom_natCast p v hv a.val, ZMod.natCast_zmod_val]
section Vectors
variable {J G : Type*} [Fintype J] [DecidableEq J] [AddCommGroup G]
variable (p : J → ℕ) (v : J → G) (hv : ∀ j, p j • v j = 0)
def vectorHom : (∀ j, ZMod (p j)) →+ G where
toFun a := ∑ j, torsionHom (p j) (v j) (hv j) (a j)
map_zero' := by simp
map_add' a b := by simp [map_add, Finset.sum_add_distrib]
lemma vectorHom_single (i : J) (a : ZMod (p i)) :
vectorHom p v hv (Pi.single i a) = torsionHom (p i) (v i) (hv i) a := by
classical
simp only [vectorHom, AddMonoidHom.coe_mk, ZeroHom.coe_mk]
rw [Finset.sum_eq_single i]
· simp
· intro j _ hj
simp [Pi.single_eq_of_ne hj]
· simp
omit [DecidableEq J] in
lemma vectorHom_one : vectorHom p v hv 1 = ∑ j, v j := by
change (∑ j, torsionHom (p j) (v j) (hv j) 1) = _
apply Finset.sum_congr rfl
intro j _
simpa using torsionHom_natCast (p j) (v j) (hv j) 1
variable [Fintype G] [∀ j, NeZero (p j)]
include hv
theorem vector_displacement
(hcop : Pairwise (fun i j => (p i).Coprime (p j)))
(B : ℝ) (hB : 1 ≤ B) (hpB : ∀ j, (p j : ℝ) ≤ B)
(h : ℕ) (hh : 0 < h) (H : G → EditCuts.RealL1) (i : J) (u : ZMod (p i)) :
variation H (u.val • v i) ≤ B ^ (2 * h) * variation H (∑ j, v j) +
(2 / (h : ℝ)) * ∑ j, avg (fun a : ZMod (p j) => variation H (a.val • v j)) := by
simpa only [vectorHom_single, vectorHom_one, torsionHom_apply] using
prime_product_displacement p hcop B hB hpB h hh (vectorHom p v hv) H i u
theorem prime_vector_displacement (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) (H : G → EditCuts.RealL1) (i : J) (u : ZMod (p i)) :
variation H (u.val • v i) ≤ (2 * P) ^ (2 * h) * variation H (∑ j, v j) +
(2 / (h : ℝ)) * ∑ j, avg (fun a : ZMod (p j) => variation H (a.val • v j)) := by
apply vector_displacement p v hv (fun i j hij => ?_) (2 * P) hP hpP h hh H i u
exact (hp i).coprime_iff_not_dvd.mpr (fun hd =>
hij (hinj ((Nat.dvd_prime (hp j)).mp hd |>.resolve_left (hp i).ne_one)))
end Vectors
end EditFourier
end OAI
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