File size: 16,021 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 | import OAI.Combinatorics.FiniteCircle.CircleWords
namespace OAI
namespace FiniteCircle
open scoped BigOperators List
abbrev MaskResidues (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) := (k : Fin h) → ZMod (p l k)
noncomputable def maskPhaseHom (h : ℕ) (p : ℕ → ℕ → ℕ) (M l : ℕ)
(hd : ∀ k : Fin h, p l k ∣ M) : MaskResidues h p l →+ (MaskTag h p l → ZMod M) where
toFun z j := (∑ k, subcircle (hd k) (z k)) + j.2.val • subcircle (hd j.1) (z j.1)
map_zero' := by funext j; simp
map_add' z u := by
funext j
simp only [Pi.add_apply, map_add, Finset.sum_add_distrib, nsmul_add]
abel
theorem maskPhaseHom_natCast (h : ℕ) (p : ℕ → ℕ → ℕ) (M l a : ℕ)
(hd : ∀ k : Fin h, p l k ∣ M) (hn : ∀ k : Fin h, 0 < p l k) :
maskPhaseHom h p M l hd (fun k => (a : ZMod (p l k))) = maskGamma h p M l a := by
funext j
change (∑ k, subcircle (hd k) (a : ZMod (p l k))) +
j.2.val • subcircle (hd j.1) (a : ZMod (p l j.1)) = _
have he (k : Fin h) : subcircle (hd k) (a : ZMod (p l k)) =
((M/p l k*(a%p l k) : ℕ) : ZMod M) := by
let : NeZero (p l k) := ⟨(hn k).ne'⟩
rw [subcircle_val, ZMod.val_natCast]
simp_rw [he]
simp only [maskGamma, nsmul_eq_mul, Nat.cast_mul]
ring
theorem sum_fin_zmod {p : ℕ} [NeZero p] (g : ZMod p → ℝ) :
∑ b : Fin p, g (b.val : ZMod p) = ∑ b : ZMod p, g b := by
refine Fintype.sum_equiv (ZMod.finEquiv p).toEquiv _ _ ?_
intro b
congr 1
cases p with
| zero => exact Fin.elim0 b
| succ p =>
change (b.val : ZMod (p+1)) = b
exact ZMod.natCast_zmod_val (show ZMod (p+1) from b)
noncomputable def maskCost (h : ℕ) (p : ℕ → ℕ → ℕ) (M : ℕ)
(hd : ∀ l (k : Fin h), p l k ∣ M) (hn : ∀ l (k : Fin h), 0 < p l k) :
ℕ → PhaseCost (ZMod M)
| 0 => circlePhaseCost M
| l+1 => by
let : ∀ k : Fin h, NeZero (p (l+1) k) := fun k => ⟨(hn _ k).ne'⟩
exact (maskCost h p M hd hn l).step (maskPhaseHom h p M (l+1) (hd _))
theorem maskCost_le_half {h M : ℕ} {p : ℕ → ℕ → ℕ} [NeZero M]
(hh : 0 < h) (hd : ∀ l (k : Fin h), p l k ∣ M) (hn : ∀ l (k : Fin h), 0 < p l k)
(l : ℕ) (x : ZMod M) : maskCost h p M hd hn l x ≤ 1/2 := by
induction l with
| zero => exact circleCost_le_half x
| succ l ih =>
let : ∀ k : Fin h, NeZero (p (l+1) k) := fun k => ⟨(hn _ k).ne'⟩
let : Nonempty (MaskTag h p (l+1)) := ⟨⟨⟨0,hh⟩,⟨0,hn _ ⟨0,hh⟩⟩⟩⟩
exact ((maskCost h p M hd hn l).infAverage_le_self _ x).trans ih
theorem card_maskTag (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) :
Fintype.card (MaskTag h p l) = ∑ k : Fin h, p l k := by
simp only [MaskTag, Fintype.card_sigma, Fintype.card_fin]
theorem mask_average_formula {h M l : ℕ} {p : ℕ → ℕ → ℕ}
(hd : ∀ k : Fin h, p l k ∣ M) (hn : ∀ k : Fin h, 0 < p l k)
(f : PhaseCost (ZMod M)) (x : ZMod M) (z : MaskResidues h p l) :
f.average (maskPhaseHom h p M l hd) x z =
(∑ k : Fin h, ∑ b : Fin (p l k),
f (x+(∑ k,subcircle (hd k) (z k))+b.val • subcircle (hd k) (z k))) /
(∑ k : Fin h, (p l k : ℝ)) := by
let : ∀ k : Fin h, NeZero (p l k) := fun k => ⟨(hn k).ne'⟩
simp only [PhaseCost.average, card_maskTag, Nat.cast_sum, Fintype.sum_sigma]
congr 1
apply Finset.sum_congr rfl
intro k _
apply Finset.sum_congr rfl
intro b _
congr 1
change x+((∑ k,subcircle (hd k) (z k))+b.val • subcircle (hd k) (z k)) = _
abel
theorem mask_active_average_lower {h M l : ℕ} {p : ℕ → ℕ → ℕ}
(hh : 0 < h) (hd : ∀ k : Fin h, p l k ∣ M)
(hp : ∀ k : Fin h, (p l k).Prime)
(hlo : ∀ k : Fin h, h^3 ≤ p l k) (hhi : ∀ k : Fin h, p l k ≤ 2*h^3)
(f : PhaseCost (ZMod M)) (x : ZMod M) (z : MaskResidues h p l)
(Q : ℝ) (hQ : 0 ≤ Q)
(hq : ∀ k : Fin h, z k ≠ 0 → Q ≤ f (subcircle (hd k) 1)) :
(((Finset.univ.filter (fun k : Fin h => z k ≠ 0)).card : ℝ)/(4*h))*Q ≤
f.average (maskPhaseHom h p M l hd) x z := by
classical
let : ∀ k : Fin h, NeZero (p l k) := fun k => ⟨(hp k).ne_zero⟩
let S := Finset.univ.filter (fun k : Fin h => z k ≠ 0)
let T : ℝ := ∑ k : Fin h, (p l k : ℝ)
let B (k : Fin h) : ℝ := ∑ b : Fin (p l k),
f (x+(∑ j,subcircle (hd j) (z j))+b.val • subcircle (hd k) (z k))
have hB (k : Fin h) : 0 ≤ B k := Finset.sum_nonneg (fun _ _ => f.nonneg _)
have hT : 0 < T := by
apply Finset.sum_pos
· intro k _; exact_mod_cast (hp k).pos
· exact Finset.univ_nonempty_iff.mpr ⟨⟨0,hh⟩⟩
have hTu : T ≤ 2*(h:ℝ)^4 := by
calc
T ≤ ∑ _k : Fin h, (2:ℝ)*h^3 := Finset.sum_le_sum (fun k _ => by exact_mod_cast hhi k)
_ = _ := by simp; ring
have hkb (k : Fin h) (hk : k ∈ S) : (h:ℝ)^3*Q ≤ 2*B k := by
have ha := f.prime_grid_bound (hp k) (subcircle (hd k))
(x+∑ j,subcircle (hd j) (z j)) (z k) ((Finset.mem_filter.mp hk).2)
have he : B k = ∑ b : ZMod (p l k),
f (x+(∑ j,subcircle (hd j) (z j))+b.val • subcircle (hd k) (z k)) := by
symm
rw [← sum_fin_zmod]
apply Finset.sum_congr rfl
intro b _
rw [ZMod.val_natCast, Nat.mod_eq_of_lt b.isLt]
rw [← he] at ha
calc
(h:ℝ)^3*Q ≤ (p l k : ℝ)*Q := mul_le_mul_of_nonneg_right (by exact_mod_cast hlo k) hQ
_ ≤ (p l k : ℝ)*f (subcircle (hd k) 1) := mul_le_mul_of_nonneg_left (hq k ((Finset.mem_filter.mp hk).2)) (by positivity)
_ ≤ _ := ha
have hsum : (S.card : ℝ)*(h:ℝ)^3*Q ≤ 2*∑ k : Fin h, B k := by
calc
_ = ∑ _k ∈ S, (h:ℝ)^3*Q := by simp; ring
_ ≤ ∑ k ∈ S, 2*B k := Finset.sum_le_sum hkb
_ ≤ ∑ k : Fin h, 2*B k := Finset.sum_le_sum_of_subset_of_nonneg
(Finset.subset_univ _) (fun k _ _ => mul_nonneg (by norm_num) (hB k))
_ = _ := (Finset.mul_sum ..).symm
rw [mask_average_formula hd (fun k => (hp k).pos)]
change (S.card : ℝ)/(4*h)*Q ≤ (∑ k : Fin h, B k)/T
rw [le_div_iff₀ hT]
have hh' : (0:ℝ) < h := by exact_mod_cast hh
have hb : 0 ≤ (S.card : ℝ)/(4*h)*Q := by positivity
calc
_ ≤ ((S.card : ℝ)/(4*h)*Q)*(2*(h:ℝ)^4) := mul_le_mul_of_nonneg_left hTu hb
_ = ((S.card : ℝ)*(h:ℝ)^3*Q)/2 := by field_simp; ring
_ ≤ _ := by linarith
theorem mask_inactive_average_lower {h M l : ℕ} {p : ℕ → ℕ → ℕ}
(hh : 0 < h) (hd : ∀ k : Fin h, p l k ∣ M)
(hp : ∀ k : Fin h, (p l k).Prime)
(hlo : ∀ k : Fin h, h^3 ≤ p l k) (hhi : ∀ k : Fin h, p l k ≤ 2*h^3)
(f : PhaseCost (ZMod M)) (x : ZMod M) (z : MaskResidues h p l)
(hs : 2*(Finset.univ.filter (fun k : Fin h => z k ≠ 0)).card ≤ h) :
(1/4:ℝ)*f (x+∑ k,subcircle (hd k) (z k)) ≤
f.average (maskPhaseHom h p M l hd) x z := by
classical
let N := Finset.univ.filter (fun k : Fin h => z k = 0)
let T : ℝ := ∑ k : Fin h, (p l k : ℝ)
let B (k : Fin h) : ℝ := ∑ b : Fin (p l k),
f (x+(∑ j,subcircle (hd j) (z j))+b.val • subcircle (hd k) (z k))
let Q := f (x+∑ k,subcircle (hd k) (z k))
have hQ : 0 ≤ Q := f.nonneg _
have hB (k : Fin h) : 0 ≤ B k := Finset.sum_nonneg (fun _ _ => f.nonneg _)
have hT : 0 < T := by
apply Finset.sum_pos
· intro k _; exact_mod_cast (hp k).pos
· exact Finset.univ_nonempty_iff.mpr ⟨⟨0,hh⟩⟩
have hTu : T ≤ 2*(h:ℝ)^4 := by
calc
T ≤ ∑ _k : Fin h, (2:ℝ)*h^3 := Finset.sum_le_sum (fun k _ => by exact_mod_cast hhi k)
_ = _ := by simp; ring
have hN : (h:ℝ) ≤ 2*N.card := by
have hc := Finset.card_filter_add_card_filter_not (s := Finset.univ) (fun k : Fin h => z k = 0)
simp only [Finset.card_univ, Fintype.card_fin] at hc
change N.card + (Finset.univ.filter (fun k : Fin h => z k ≠ 0)).card = h at hc
exact_mod_cast (show h ≤ 2*N.card by omega)
have hkb (k : Fin h) (hk : k ∈ N) : (h:ℝ)^3*Q ≤ B k := by
have hz := (Finset.mem_filter.mp hk).2
have he : B k = (p l k : ℝ)*Q := by
simp [B, Q, hz]
rw [he]
exact mul_le_mul_of_nonneg_right (by exact_mod_cast hlo k) hQ
have hsum : (N.card : ℝ)*(h:ℝ)^3*Q ≤ ∑ k : Fin h, B k := by
calc
_ = ∑ _k ∈ N, (h:ℝ)^3*Q := by simp; ring
_ ≤ ∑ k ∈ N, B k := Finset.sum_le_sum hkb
_ ≤ ∑ k : Fin h, B k := Finset.sum_le_sum_of_subset_of_nonneg
(Finset.subset_univ _) (fun k _ _ => hB k)
rw [mask_average_formula hd (fun k => (hp k).pos)]
change (1/4:ℝ)*Q ≤ (∑ k : Fin h, B k)/T
rw [le_div_iff₀ hT]
calc
_ ≤ ((1/4:ℝ)*Q)*(2*(h:ℝ)^4) := mul_le_mul_of_nonneg_left hTu (by positivity)
_ = ((h:ℝ)/2)*h^3*Q := by ring
_ ≤ (N.card : ℝ)*h^3*Q := by gcongr; linarith
_ ≤ _ := hsum
theorem mask_minimizer_options {h M l : ℕ} {p : ℕ → ℕ → ℕ}
[∀ k : Fin h, NeZero (p l k)]
(hh : 0 < h) (hd : ∀ k : Fin h, p l k ∣ M)
(hp : ∀ k : Fin h, (p l k).Prime)
(hlo : ∀ k : Fin h, h^3 ≤ p l k) (hhi : ∀ k : Fin h, p l k ≤ 2*h^3)
(f : PhaseCost (ZMod M)) (x : ZMod M) :
∃ z : MaskResidues h p l,
(let S := Finset.univ.filter (fun k : Fin h => z k ≠ 0)
(∀ Q : ℝ, 0 ≤ Q → (∀ k ∈ S, Q ≤ f (subcircle (hd k) 1)) →
(S.card : ℝ)/(4*h)*Q ≤ f.infAverage (maskPhaseHom h p M l hd) x) ∧
(2*S.card ≤ h → (1/4:ℝ)*f (x+∑ k,subcircle (hd k) (z k)) ≤
f.infAverage (maskPhaseHom h p M l hd) x)) := by
classical
let : ∀ k : Fin h, NeZero (p l k) := fun k => ⟨(hp k).ne_zero⟩
obtain ⟨z,hz⟩ := f.exists_average_eq (maskPhaseHom h p M l hd) x
refine ⟨z, ?_⟩
dsimp only
rw [hz]
constructor
· intro Q hQ hq
exact mask_active_average_lower hh hd hp hlo hhi f x z Q hQ
(fun k hk => hq k (by simp [hk]))
· exact mask_inactive_average_lower hh hd hp hlo hhi f x z
theorem deficit_cast {β : Type*} (x y : List β) : (deficit x y : ℝ) = (x.length:ℝ) - lcs x y := by
exact Nat.cast_sub (lcs_le_length_left x y)
theorem deficit_taggedRow_lower {J β : Type*} [Fintype J] [DecidableEq J]
(js : List J) (hn : js.Nodup) (hall : ∀ j, j ∈ js) (W U : J → List β) :
(∑ j : J, (deficit (W j) (U j) : ℝ)) ≤
deficit (taggedRow js W) (taggedRow js U) := by
have hc := lcs_taggedRow_le js hn hall W U
have hcr : (lcs (taggedRow js W) (taggedRow js U) : ℝ) ≤ ∑ j : J, (lcs (W j) (U j) : ℝ) := by exact_mod_cast hc
simp_rw [deficit_cast]
rw [Finset.sum_sub_distrib, length_taggedRow js hn hall W, Nat.cast_sum]
linarith
theorem maskLength_pos {h M R : ℕ} {p : ℕ → ℕ → ℕ}
(hh : 0 < h) (hM : 0 < M) (hR : 0 < R) (hn : ∀ l (k : Fin h), 0 < p l k)
(l : ℕ) : 0 < maskLength h p R M l := by
induction l with
| zero => exact hM
| succ l ih =>
unfold maskLength
apply Nat.mul_pos (Nat.mul_pos hR (Finset.prod_pos (fun k _ => hn _ k)))
apply Nat.mul_pos _ ih
rw [length_maskTags]
exact Finset.sum_pos (fun k _ => hn _ k) (Finset.univ_nonempty_iff.mpr ⟨⟨0,hh⟩⟩)
theorem mask_separation {h M R : ℕ} {p : ℕ → ℕ → ℕ} [NeZero M]
(he : M = 2*(M/2)) (hh : 0 < h) (hR : 0 < R)
(hd : ∀ l (k : Fin h), p l k ∣ M) (hn : ∀ l (k : Fin h), 0 < p l k)
(ht : ∀ l, 4 ≤ (maskTags h p l).length)
(l : ℕ) (v : MaskState h p M l) (θ : ZMod M) :
(1/((2:ℝ)^l*4))*(maskLength h p R M l)*maskCost h p M hd hn l θ ≤
(deficit (maskWord h p R M l v)
(maskWord h p R M l (shiftMaskState h p M l v θ)) : ℝ) := by
classical
induction l generalizing θ with
| zero =>
change ZMod M at v
change (1/((2:ℝ)^0*4))*M*circleCost θ ≤
(deficit (maskWord h p R M 0 v) (maskWord h p R M 0 (v+θ)) : ℝ)
rw [maskWord_zero_eq, maskWord_zero_eq]
calc
_ = (M:ℝ)*circleCost θ/4 := by norm_num; ring
_ ≤ _ := halfCircle_separation he v θ
| succ l ih =>
let : ∀ k : Fin h, NeZero (p (l+1) k) := fun k => ⟨(hn _ k).ne'⟩
let L := maskLength h p R M l
let t := (maskTags h p (l+1)).length
let N := R*period h p (l+1)
let f := maskCost h p M hd hn l
let σ := maskPhaseHom h p M (l+1) (hd _)
let ρ := (1/((2:ℝ)^l*4))*maskCost h p M hd hn (l+1) θ
have hL : 0 < L := maskLength_pos hh (NeZero.pos M) hR hn l
have hp2 : 1 ≤ (2:ℝ)^l := one_le_pow₀ (by norm_num)
have hρ : ρ ≤ 1/2 := by
have hc := maskCost_le_half hh hd hn (l+1) θ
have hc0 := (maskCost h p M hd hn (l+1)).nonneg θ
dsimp [ρ]
have hi : 1/((2:ℝ)^l*4) ≤ 1 := (div_le_one (by positivity)).mpr (by nlinarith)
calc
_ ≤ 1*maskCost h p M hd hn (l+1) θ := mul_le_mul_of_nonneg_right hi hc0
_ ≤ _ := by simpa using hc
have hx : (maskWord h p R M (l+1) v).length = (t*L)*N := by
rw [length_maskWord]
change N*(t*L) = (t*L)*N
exact Nat.mul_comm _ _
have hy : (maskWord h p R M (l+1) (shiftMaskState h p M (l+1) v θ)).length = (t*L)*N := by
rw [length_maskWord]
change N*(t*L) = (t*L)*N
exact Nat.mul_comm _ _
have hr := row_separation hL (ht (l+1)) hx hy
(fun x => (maskTags h p (l+1)).idxOf x.1)
(tag_get_maskWord h p R M l v hL (by have := ht (l+1); omega))
(tag_get_maskWord h p R M l _ hL (by have := ht (l+1); omega)) ρ hρ
have hrows : ∀ i j : Fin N, ρ*(t*L) ≤
(deficit (segment (maskWord h p R M (l+1) v) (i.val*(t*L)) (t*L))
(segment (maskWord h p R M (l+1) (shiftMaskState h p M (l+1) v θ))
(j.val*(t*L)) (t*L)) : ℝ) := by
intro a b
rw [segment_maskWord, segment_maskWord]
let z : MaskResidues h p (l+1) := fun k => (b.val : ZMod (p (l+1) k)) - (a.val : ZMod (p (l+1) k))
have hz (j : MaskTag h p (l+1)) : σ z j =
maskGamma h p M (l+1) b j - maskGamma h p M (l+1) a j := by
change σ ((fun k : Fin h => (b.val : ZMod (p (l+1) k))) - (fun k : Fin h => (a.val : ZMod (p (l+1) k)))) j = _
rw [map_sub, Pi.sub_apply]
exact congrArg₂ (·-·) (congrFun (maskPhaseHom_natCast h p M (l+1) b (hd _) (hn _)) j)
(congrFun (maskPhaseHom_natCast h p M (l+1) a (hd _) (hn _)) j)
have hind (j : MaskTag h p (l+1)) := ih
(shiftMaskState h p M l (v j) (maskGamma h p M (l+1) a j)) (θ+σ z j)
have hstate (j : MaskTag h p (l+1)) :
shiftMaskState h p M l (shiftMaskState h p M l (v j) (maskGamma h p M (l+1) a j)) (θ+σ z j) =
shiftMaskState h p M l (shiftMaskState h p M l (v j) θ) (maskGamma h p M (l+1) b j) := by
rw [shiftMaskState_add, shiftMaskState_add, hz]
congr 1
abel
simp_rw [hstate] at hind
have hsum := Finset.sum_le_sum (s := Finset.univ) (fun j _ => hind j)
have hdef := deficit_taggedRow_lower (maskTags h p (l+1)) (nodup_maskTags ..)
(mem_maskTags h p (l+1))
(fun j => maskWord h p R M l (shiftMaskState h p M l (v j) (maskGamma h p M (l+1) a j)))
(fun j => maskWord h p R M l (shiftMaskState h p M l (shiftMaskState h p M l (v j) θ) (maskGamma h p M (l+1) b j)))
apply le_trans _ (hsum.trans hdef)
rw [← Finset.mul_sum]
have hcost : maskCost h p M hd hn (l+1) θ ≤ f.average σ θ z := f.infAverage_le σ θ z
have htpos : (0:ℝ) < t := by exact_mod_cast (show 0 < t by have := ht (l+1); omega)
have hcard : (Fintype.card (MaskTag h p (l+1)) : ℝ) = t := by rw [card_maskTag]; exact_mod_cast (length_maskTags h p (l+1)).symm
rw [PhaseCost.average, hcard, le_div_iff₀ htpos] at hcost
dsimp only [ρ]
calc
_ = ((1/((2:ℝ)^l*4))*(L:ℝ)) * ((maskCost h p M hd hn (l+1) θ)*t) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hcost (by positivity)
have hfinal := hr hrows
rw [length_maskWord] at hfinal
dsimp only [ρ] at hfinal
calc
_ = ((1/((2:ℝ)^l*4))*maskCost h p M hd hn (l+1) θ)/2 * maskLength h p R M (l+1) := by rw [pow_succ]; ring
_ ≤ _ := hfinal
end FiniteCircle
end OAI
|