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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 | import OAI.Combinatorics.EditDistance.LowerBound
import OAI.Combinatorics.EditDistance.DistortionMonotonicity
namespace OAI
/-!
# Uniform lower bounds under binary alphabet embeddings
-/
namespace EditDistortion
universe u
lemma coordinate_norm {X : Type u} [Fintype X] (x : X) : ‖coordinateEmbedding X x‖ = 1 := by
classical
change ‖lp.single (E := fun _ : ℕ => ℝ) 1 ((Fintype.equivFin X x).val) (1 : ℝ)‖ = 1
rw [lp.norm_single (by norm_num)]
norm_num
lemma coordinate_separation {X : Type u} [Fintype X] (x y : X) (hxy : x ≠ y) :
1 ≤ ‖coordinateEmbedding X x-coordinateEmbedding X y‖ := by
classical
let e : X ↪ ℕ := (Fintype.equivFin X).toEmbedding.trans ⟨Fin.val,Fin.val_injective⟩
have hne : e x ≠ e y := fun h => hxy (e.injective h)
have h := lp.norm_apply_le_norm (by norm_num : (1 : ENNReal) ≠ 0)
(coordinateEmbedding X x - coordinateEmbedding X y) (e x)
have he : (coordinateEmbedding X x-coordinateEmbedding X y) (e x) = 1 := by
change lp.single (E := fun _ : ℕ => ℝ) 1 (e x) (1 : ℝ) (e x) -
lp.single (E := fun _ : ℕ => ℝ) 1 (e y) (1 : ℝ) (e x) = 1
rw [lp.single_apply_self,lp.single_apply_ne (E := fun _ : ℕ => ℝ) 1 (e y) (1 : ℝ) hne]
norm_num
simpa only [he,norm_one] using h
theorem alphabet_coarse_upper (Alpha : Type u) [Fintype Alpha] [Nonempty Alpha]
{d : ℕ} (hd : 1 ≤ d) : E Alpha d ≤ 2*(d : ℝ) := by
classical
let : Finite (Words Alpha d) := finite_words Alpha d
let : Fintype (Words Alpha d) := Fintype.ofFinite _
let x : Words Alpha d := ⟨[],by simp⟩
let y : Words Alpha d := ⟨[Classical.arbitrary Alpha],by simpa using hd⟩
let : Nontrivial (Words Alpha d) := ⟨⟨x,y,by simp [x,y]⟩⟩
let f := coordinateEmbedding (Words Alpha d)
apply (leastDistortion_le _ (fun v => edit_real_diag v.val)
(fun v w h => edit_real_pos v.val w.val (fun hh => h (Subtype.ext hh))) f).trans
apply distortion_le_of_bounds _
(fun v w h => edit_real_pos v.val w.val (fun hh => h (Subtype.ext hh))) f (by norm_num)
· intro v w
by_cases h : v = w
· subst w
simp [edit_real_diag]
· have hp : (1 : ℝ) ≤ edit v.val w.val := by
exact_mod_cast Nat.one_le_iff_ne_zero.mpr
(mt (cost_eq_zero_iff (s := true)).mp (fun hh => h (Subtype.ext hh)))
have hh := norm_sub_le (f v) (f w)
have hv : ‖f v‖ = 1 := coordinate_norm v
have hw : ‖f w‖ = 1 := coordinate_norm w
nlinarith
· intro v w
by_cases h : v = w
· subst w
simp [edit_real_diag]
· have hs : (1 : ℝ) ≤ ‖f v-f w‖ := coordinate_separation v w h
have he : (edit v.val w.val : ℝ) ≤ d := by
exact_mod_cast (edit_le_max_length v.val w.val).trans
(max_le v.property w.property)
nlinarith [Nat.cast_nonneg (α := ℝ) d]
lemma alphabet_values_bddAbove {d : ℕ} (hd : 1 ≤ d) :
BddAbove {r : ℝ | ∃ n : ℕ, 2 ≤ n ∧ r = E (Fin n) d} := by
refine ⟨2*(d : ℝ),?_⟩
rintro r ⟨n,hn,rfl⟩
let : NeZero n := ⟨by omega⟩
exact alphabet_coarse_upper (Fin n) hd
lemma alphabet_le_sup {d n : ℕ} (hd : 1 ≤ d) (hn : 2 ≤ n) : E (Fin n) d ≤ alphabetSup d :=
le_csSup (alphabet_values_bddAbove hd) ⟨n,hn,rfl⟩
end EditDistortion
namespace EditLower
universe u
open EditDistortion
theorem all_lower_clauses : ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →
(∀ (Alpha : Type u) [Fintype Alpha], 2 ≤ Fintype.card Alpha →
exponentScale c d ≤ E Alpha d) ∧ BinaryWitness c d ∧
exponentScale c d ≤ alphabetSup d := by
obtain ⟨D,hD⟩ := Filter.eventually_atTop.mp eventually_length_witness
refine ⟨1/1200,by norm_num,max D 1,?_⟩
intro d hd
obtain ⟨n,hn,S,hfin,hne,hnt,hlen,hs⟩ := hD d ((le_max_left _ _).trans hd)
refine ⟨?_,⟨n,hn,S,hfin,hne,hlen,hs⟩,?_⟩
· intro Alpha inst hAlpha
exact hs.trans (binary_subset_lower Alpha hAlpha hn S hfin hnt hlen)
· exact (hs.trans (binary_subset_lower (Fin 2) (by simp) hn S hfin hnt hlen)).trans
(alphabet_le_sup ((le_max_right _ _).trans hd) (by omega))
end EditLower
end OAI
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