| 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 |
| |
