| import OAI.Combinatorics.EditDistance.CutModels |
| import OAI.Combinatorics.EditDistance.DirectSums |
|
|
| namespace OAI |
|
|
| /-! |
| # Realization of all finite-cut distances by one summable vector map |
| -/ |
|
|
| namespace EditCutModel |
| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| open Finset |
|
|
| noncomputable def Model.vector {X : Type*} (M : Model X) (x : X) : EditL1.L1 := |
| EditL1.pack (fun j : Option M.Index => match j with |
| | none => 0 |
| | some i => lp.single (E := fun _ : ℕ => ℝ) 1 0 (if M.cut i x then M.weight i else 0)) |
|
|
| lemma singleton_sub_norm (a b : ℝ) : |
| ‖lp.single (E := fun _ : ℕ => ℝ) 1 0 a - lp.single (E := fun _ : ℕ => ℝ) 1 0 b‖ = |a-b| := by |
| have he : lp.single (E := fun _ : ℕ => ℝ) 1 0 a - lp.single (E := fun _ : ℕ => ℝ) 1 0 b = |
| lp.single (E := fun _ : ℕ => ℝ) 1 0 (a-b) := by |
| ext i |
| change (lp.single (E := fun _ : ℕ => ℝ) 1 0 a) i - (lp.single (E := fun _ : ℕ => ℝ) 1 0 b) i = (lp.single (E := fun _ : ℕ => ℝ) 1 0 (a-b)) i |
| by_cases hi : i = 0 |
| · subst i |
| simp only [lp.single_apply_self] |
| · rw [lp.single_apply_ne (E := fun _ : ℕ => ℝ) 1 0 a hi, |
| lp.single_apply_ne (E := fun _ : ℕ => ℝ) 1 0 b hi, |
| lp.single_apply_ne (E := fun _ : ℕ => ℝ) 1 0 (a-b) hi,sub_self] |
| rw [he,lp.norm_single (by norm_num)] |
| exact Real.norm_eq_abs _ |
|
|
| lemma Model.vector_distance {X : Type*} (M : Model X) (x y : X) : |
| ‖M.vector x-M.vector y‖ = M.dist x y := by |
| unfold Model.vector |
| rw [EditL1.pack_distance,Fintype.sum_option] |
| simp only [sub_self,norm_zero,zero_add] |
| unfold Model.dist |
| apply sum_congr rfl |
| intro i _ |
| rw [singleton_sub_norm] |
| cases hix : M.cut i x <;> cases hiy : M.cut i y <;> |
| simp [sep, abs_of_nonneg (M.nonneg i)] |
|
|
| end EditCutModel |
|
|
| end OAI |
|
|
