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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 | 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
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