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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 | import OAI.Combinatorics.EditDistance.BinaryEmbedding
import OAI.Combinatorics.EditDistance.Distortion
import OAI.Combinatorics.EditDistance.Families
namespace OAI
/-!
# Distortion lower bounds from separated finite word families
-/
namespace EditLower
open EditDistortion EditFamily EditCode EditAverage EditFourier
variable {p n A t : ℕ} {α Q : ℝ} (F : Family p n A α Q)
(label : Fin A ↪ (Fin t → Bool))
def codeWord (z : F.State) : List Bool := encode label (List.ofFn (F.word z))
def codeSet : Set (List Bool) := Set.range (codeWord F label)
def codePoint (z : F.State) : codeSet F label := ⟨codeWord F label z,⟨z,rfl⟩⟩
lemma codeSet_finite : (codeSet F label).Finite := Set.finite_range _
lemma codeSet_nonempty : (codeSet F label).Nonempty := ⟨_,⟨0,rfl⟩⟩
lemma codeSet_length (x : List Bool) (hx : x ∈ codeSet F label) : x.length = width t*n := by
obtain ⟨z,rfl⟩ := hx
simp only [codeWord,encode_length,List.length_ofFn]
lemma far_after_code (hp : 1 < p) (z : F.State) :
(n : ℝ)*α ≤ 4*(edit (codeWord F label z) (codeWord F label (z+F.v)) : ℝ) := by
have hf := F.far z 1 (by omega) hp
simp only [one_nsmul] at hf
have hc := indel_code_lower label label.injective (List.ofFn (F.word z))
(List.ofFn (F.word (z+F.v))) 0 0
simp only [padded,List.replicate_zero,List.append_nil] at hc
have he := indel_le_two_edit (codeWord F label z) (codeWord F label (z+F.v))
have hcR : (indel (List.ofFn (F.word z)) (List.ofFn (F.word (z+F.v))) : ℝ) ≤
2*(indel (codeWord F label z) (codeWord F label (z+F.v)) : ℝ) := by exact_mod_cast hc
have heR : (indel (codeWord F label z) (codeWord F label (z+F.v)) : ℝ) ≤
2*(edit (codeWord F label z) (codeWord F label (z+F.v)) : ℝ) := by exact_mod_cast he
linarith
lemma codeSet_nontrivial (hp : 1 < p) (hn : 0 < n) (ha : 0 < α) : Nontrivial (codeSet F label) := by
have h := far_after_code F label hp 0
refine ⟨⟨codePoint F label 0,codePoint F label (0+F.v),?_⟩⟩
intro he
have he' : codeWord F label 0 = codeWord F label (0+F.v) := congrArg Subtype.val he
rw [←he',edit_real_diag] at h
have : (0 : ℝ) < n := by exact_mod_cast hn
nlinarith
theorem family_distortion (hp : 1 < p) (hn : 0 < n) (ha : 0 < α)
(f : codeSet F label ↪ L1) :
α ≤ 4*(width t : ℝ)*Q*distortion (fun x y : codeSet F label => (edit x.val y.val : ℝ)) f := by
classical
let : Fintype (codeSet F label) := (codeSet_finite F label).fintype
let : Nontrivial (codeSet F label) := codeSet_nontrivial F label hp hn ha
let ρ : codeSet F label → codeSet F label → ℝ := fun x y => (edit x.val y.val : ℝ)
obtain ⟨U,V,hU,hV,hD,hu,hv⟩ := distortion_constants ρ
(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
let C : ℝ := U*(width t : ℝ)*n
have hnR : (0 : ℝ) < n := by exact_mod_cast hn
have hwR : (0 : ℝ) < width t := by exact_mod_cast EditCode.width_pos (t := t)
have hC : 0 < C := by dsimp [C]; positivity
let H : F.State → L1 := fun z => C⁻¹ • f (codePoint F label z)
have hnorm (z e : F.State) : ‖H (z+e)-H z‖ =
‖f (codePoint F label (z+e))-f (codePoint F label z)‖/C := by
simp only [H,←smul_sub,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hC),div_eq_mul_inv,mul_comm]
have hnorm' (z e : F.State) : ‖f (codePoint F label (z+e))-f (codePoint F label z)‖ =
C*‖H (z+e)-H z‖ := by rw [hnorm]; field_simp
have hcheap (e : F.State) (r : ℝ) (hr : (e,r) ∈ F.steps) : variation H e ≤ r := by
unfold variation
rw [←avg_const (X := F.State) r]
apply avg_mono
intro z
rw [hnorm,div_le_iff₀ hC,norm_sub_rev]
have h1 := hu (codePoint F label z) (codePoint F label (z+e))
have h2 := edit_le_indel (codeWord F label z) (codeWord F label (z+e))
have h3 := indel_code_upper label (List.ofFn (F.word z)) (List.ofFn (F.word (z+e)))
have h23 : (edit (codeWord F label z) (codeWord F label (z+e)) : ℝ) ≤
(width t : ℝ)*(indel (List.ofFn (F.word z)) (List.ofFn (F.word (z+e))) : ℝ) := by
exact_mod_cast h2.trans h3
have h4 := F.cheap e r hr z
change ‖f (codePoint F label z)-f (codePoint F label (z+e))‖ ≤
U*(edit (codeWord F label z) (codeWord F label (z+e)) : ℝ) at h1
calc
_ ≤ U*((width t : ℝ)*((n : ℝ)*r)) := h1.trans (mul_le_mul_of_nonneg_left
(h23.trans (mul_le_mul_of_nonneg_left h4 hwR.le)) hU.le)
_ = r*C := by dsimp [C]; ring
have hbound := F.control H hcheap 1 (by omega) hp
simp only [one_nsmul] at hbound
have hfar (z : F.State) : (n : ℝ)*α ≤ 4*V*C*‖H (z+F.v)-H z‖ := by
have h1 := far_after_code F label hp z
have h2 := hv (codePoint F label z) (codePoint F label (z+F.v))
change (edit (codeWord F label z) (codeWord F label (z+F.v)) : ℝ) ≤
V*‖f (codePoint F label z)-f (codePoint F label (z+F.v))‖ at h2
rw [norm_sub_rev,hnorm'] at h2
nlinarith
have havg : (n : ℝ)*α ≤ 4*V*C*variation H F.v := by
change (n : ℝ)*α ≤ 4*V*C*avg (fun z => ‖H (z+F.v)-H z‖)
rw [←avg_mul,←avg_const (X := F.State) ((n : ℝ)*α)]
exact avg_mono hfar
have hfinal : (n : ℝ)*α ≤ (n : ℝ)*(4*(width t : ℝ)*Q*(U*V)) := by
calc
_ ≤ 4*V*C*Q := havg.trans (mul_le_mul_of_nonneg_left hbound (by positivity))
_ = _ := by dsimp [C]; ring
have hcancel := (mul_le_mul_iff_right₀ hnR).mp hfinal
change α ≤ 4*(width t : ℝ)*Q*distortion ρ f
rw [hD]
exact hcancel
end EditLower
end OAI
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