File size: 3,808 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 | import OAI.Combinatorics.EditDistance.LowerParameters
import OAI.Combinatorics.EditDistance.LowerDistortion
namespace OAI
/-!
# Finite nonempty binary witnesses of a common length
-/
namespace EditLower
open EditDistortion EditFamily EditCode EditParameters
noncomputable def binaryLabels (k : ℕ) : Fin (familyAlphabet (b k) (P k) k) ↪ (Fin (codeBits k) → Bool) := by
simpa only [codeBits,Fintype.card_fin] using labels (Fin (familyAlphabet (b k) (P k) k))
theorem family_ratio {k p : ℕ} (hk : 0 < k) (hp : 1 < p)
(F : Family p (familyLength (b k) (P k) k) (familyAlphabet (b k) (P k) k)
(familyAlpha k) (familyBudget (b k) (P k) (h k) k))
(f : codeSet F (binaryLabels k) ↪ L1) :
lowerRatio k ≤ distortion (fun x y : codeSet F (binaryLabels k) => (edit x.val y.val : ℝ)) f := by
classical
let label := binaryLabels k
let : Fintype (codeSet F label) := (codeSet_finite F label).fintype
have hn := familyLength_pos (by dsimp [b]; omega : 0 < b k) (prime_parameter_pos hk) k
let : Nontrivial (codeSet F label) := codeSet_nontrivial F label hp hn (familyAlpha_pos k)
let D := distortion (fun x y : codeSet F label => (edit x.val y.val : ℝ)) f
have hD : 0 ≤ D := distortion_nonneg _ (fun x => edit_real_diag x.val)
(fun x y hxy => edit_real_pos x.val y.val (fun h => hxy (Subtype.ext h))) f
have hl := family_distortion F label hp hn (familyAlpha_pos k) f
change familyAlpha k ≤ 4*(codeWidth k : ℝ)*familyBudget (b k) (P k) (h k) k*D at hl
have hw : (0 : ℝ) < codeWidth k := by exact_mod_cast EditParameters.width_pos k
have hh : (0 : ℝ) < (h k : ℝ)^k := by dsimp [h]; positivity
have hb : familyAlpha k ≤ (16*(codeWidth k : ℝ)*D)/(h k : ℝ)^k := by
calc
_ ≤ 4*(codeWidth k : ℝ)*(4/(h k : ℝ)^k)*D := by
apply hl.trans
gcongr
exact budget_small hk
_ = _ := by ring
have hmul := (le_div_iff₀ hh).mp hb
have hpow : familyAlpha k*(h k : ℝ)^k = ((h k : ℝ)/24)^k := by
unfold familyAlpha
rw [←mul_pow]
congr 1
ring
change ((h k : ℝ)/24)^k/(16*(codeWidth k : ℝ)) ≤ D
apply (div_le_iff₀ (by positivity : 0 < 16*(codeWidth k : ℝ))).mpr
rw [hpow] at hmul
nlinarith
theorem eventually_depth_witness : ∀ᶠ k : ℕ in Filter.atTop,
∃ S : Set (List Bool), S.Finite ∧ S.Nonempty ∧ S.Nontrivial ∧
(∀ w ∈ S, w.length = encodedLength k) ∧
lowerRatio k ≤ leastDistortion (fun x y : S => (edit x.val y.val : ℝ)) := by
classical
obtain ⟨B,hB⟩ := Filter.eventually_atTop.mp EditPrimes.eventually_pool_square
filter_upwards [Filter.eventually_ge_atTop (max 1 B)] with k hk
have hk0 : 0 < k := by omega
have hb0 : 0 < b k := by dsimp [b]; omega
have hpool := hB (b k) (by dsimp [b]; omega)
obtain ⟨p,hinj,hprime,hlow,hhigh⟩ := EditPrimes.primes_of_pool hpool
have hP : 1 ≤ P k := Nat.one_le_iff_ne_zero.mpr (prime_parameter_pos hk0).ne'
let j : Fin (b k) := ⟨0,hb0⟩
let F : Family (p j) (familyLength (b k) (P k) k) (familyAlphabet (b k) (P k) k)
(familyAlpha k) (familyBudget (b k) (P k) (h k) k) := by
simpa only [Fintype.card_fin] using recursive p hprime hinj (P k) hP hlow hhigh
(h k) (by dsimp [h]; omega) k j
let S := codeSet F (binaryLabels k)
let : Fintype S := (codeSet_finite F (binaryLabels k)).fintype
refine ⟨S,codeSet_finite F (binaryLabels k),codeSet_nonempty F (binaryLabels k),?_,?_,?_⟩
· exact Set.nontrivial_coe_sort.mp (codeSet_nontrivial F (binaryLabels k) (hprime j).one_lt
(familyLength_pos hb0 (prime_parameter_pos hk0) k) (familyAlpha_pos k))
· intro w hw
exact codeSet_length F (binaryLabels k) w hw
· apply leastDistortion_lower
intro f
exact family_ratio hk0 (hprime j).one_lt F f
end EditLower
end OAI
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