| 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 |
|
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| end OAI |
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