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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 | import OAI.Combinatorics.EditDistance.Distortion
import OAI.Combinatorics.EditDistance.Metric
namespace OAI
/-!
# Restriction and monotonicity of least distortion
-/
namespace EditDistortion
universe u v
lemma leastDistortion_le {X : Type u} [Finite X] [Nontrivial X]
(ρ : X → X → ℝ) (hd : ∀ x, ρ x x = 0)
(hp : ∀ x y, x ≠ y → 0 < ρ x y) (f : X ↪ L1) : leastDistortion ρ ≤ distortion ρ f := by
apply csInf_le (s := Set.range (distortion ρ)) (a := distortion ρ f) ?_ ⟨f,rfl⟩
refine ⟨0,?_⟩
rintro r ⟨g,rfl⟩
exact distortion_nonneg ρ hd hp g
lemma distortion_le_of_bounds {X : Type u} [Finite X] [Nontrivial X]
(ρ : X → X → ℝ) (hp : ∀ x y, x ≠ y → 0 < ρ x y)
(f : X ↪ L1) {U V : ℝ} (hU : 0 ≤ U)
(hu : ∀ x y, ‖f x-f y‖ ≤ U*ρ x y)
(hv : ∀ x y, ρ x y ≤ V*‖f x-f y‖) : distortion ρ f ≤ U*V := by
have hn (x y : X) (hxy : x ≠ y) : 0 < ‖f x-f y‖ :=
norm_pos_iff.mpr (sub_ne_zero.mpr (fun h => hxy (f.injective h)))
obtain ⟨x,y,hxy⟩ := exists_pair_ne X
have hn1 : {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ‖f x-f y‖/ρ x y}.Nonempty :=
⟨_,⟨x,y,hxy,rfl⟩⟩
have hn2 : {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ρ x y/‖f x-f y‖}.Nonempty :=
⟨_,⟨x,y,hxy,rfl⟩⟩
have h1 : sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ‖f x-f y‖/ρ x y} ≤ U := by
apply csSup_le hn1
rintro r ⟨i,j,hij,rfl⟩
exact (div_le_iff₀ (hp i j hij)).mpr (hu i j)
have h2 : sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ρ x y/‖f x-f y‖} ≤ V := by
apply csSup_le hn2
rintro r ⟨i,j,hij,rfl⟩
exact (div_le_iff₀ (hn i j hij)).mpr (hv i j)
have hnon : 0 ≤ sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ρ x y/‖f x-f y‖} :=
(div_nonneg (hp x y hxy).le (norm_nonneg _)).trans
(le_csSup (reverse_ratios_finite ρ f).bddAbove ⟨x,y,hxy,rfl⟩)
unfold distortion
exact mul_le_mul h1 h2 hnon hU
theorem leastDistortion_mono {X : Type u} {Y : Type v} [Fintype X] [Fintype Y]
[Nontrivial X] [Nontrivial Y] (ρX : X → X → ℝ) (ρY : Y → Y → ℝ)
(hdX : ∀ x, ρX x x = 0) (hpX : ∀ x y, x ≠ y → 0 < ρX x y)
(hdY : ∀ x, ρY x x = 0) (hpY : ∀ x y, x ≠ y → 0 < ρY x y)
(e : X ↪ Y) (he : ∀ x y, ρY (e x) (e y) = ρX x y) :
leastDistortion ρX ≤ leastDistortion ρY := by
apply leastDistortion_lower
intro f
obtain ⟨U,V,hU,_,hD,hu,hv⟩ := distortion_constants ρY hdY hpY f
let g : X ↪ L1 := e.trans f
calc
leastDistortion ρX ≤ distortion ρX g := leastDistortion_le ρX hdX hpX g
_ ≤ U*V := distortion_le_of_bounds ρX hpX g hU.le
(fun x y => by simpa only [g,Function.Embedding.trans_apply,he] using hu (e x) (e y))
(fun x y => by simpa only [g,Function.Embedding.trans_apply,he] using hv (e x) (e y))
_ = distortion ρY f := hD.symm
end EditDistortion
namespace EditDistortion
universe u
theorem binary_subset_lower (Alpha : Type u) [Fintype Alpha] (hAlpha : 2 ≤ Fintype.card Alpha)
{d n : ℕ} (hn : n ≤ d) (S : Set (List Bool)) (hfin : S.Finite) (hnt : S.Nontrivial)
(hlen : ∀ w ∈ S, w.length = n) :
leastDistortion (fun x y : S => (edit x.val y.val : ℝ)) ≤ E Alpha d := by
classical
let : Fintype S := hfin.fintype
let : Nontrivial S := hnt.coe_sort
let : Finite (Words Alpha d) := finite_words Alpha d
let : Fintype (Words Alpha d) := Fintype.ofFinite _
let a : Bool ↪ Alpha := Classical.choice (Function.Embedding.nonempty_of_card_le
(by simpa only [Fintype.card_bool] using hAlpha))
let e : S ↪ Words Alpha d :=
⟨fun x => ⟨x.val.map a,by simpa only [List.length_map,hlen x.val x.property] using hn⟩,
fun x y h => Subtype.ext ((List.map_inj_right a.injective).mp (congrArg Subtype.val h))⟩
let : Nontrivial (Words Alpha d) := by
obtain ⟨x,y,hxy⟩ := exists_pair_ne S
exact ⟨⟨e x,e y,fun h => hxy (e.injective h)⟩⟩
apply leastDistortion_mono _ _
(fun x => edit_real_diag x.val)
(fun x y hxy => edit_real_pos x.val y.val (fun h => hxy (Subtype.ext h)))
(fun x => edit_real_diag x.val)
(fun x y hxy => edit_real_pos x.val y.val (fun h => hxy (Subtype.ext h))) e
intro x y
exact_mod_cast cost_map_eq_of_leftInverse (s := true) a (Function.invFun a)
(Function.leftInverse_invFun a.injective) x.val y.val
end EditDistortion
end OAI
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