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f0f05de | 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 | import OAI.Combinatorics.BinaryEdit.CodeNumerics
import OAI.Combinatorics.BinaryEdit.EditDeficit
import OAI.Combinatorics.BinaryEdit.EncodingUpper
namespace OAI
/-! The exact arithmetic and metric estimates for fixed-length binary conversion. -/
namespace TreeEdit.BinaryLower.Code
universe u v
/-- The precise 402 deficit estimate implies the required 804 ordinary-edit
estimate for a fixed pair of original words. Both metrics are the actual
unrestricted unit-cost edit distance. -/
theorem encode_bounds_of_deficit {α : Type u} {β : Type v} {M P : ℕ}
(c : α → Word β P) {x y : List α}
(hx : x.length = M) (hy : y.length = M)
(hdef : (P : ℝ) / 402 * (deficit M x y : ℝ) ≤
(deficit (M * P) (encode c x) (encode c y) : ℝ)) :
(P : ℝ) / 804 * (ed x y : ℝ) ≤
(ed (encode c x) (encode c y) : ℝ) ∧
(ed (encode c x) (encode c y) : ℝ) ≤
(P : ℝ) * (ed x y : ℝ) := by
have hencx : (encode c x).length = M * P := by rw [length_encode, hx]
have hency : (encode c y).length = M * P := by rw [length_encode, hy]
have hsource := editDistance_deficit_bounds_real hx hy
have htarget := editDistance_deficit_bounds_real hencx hency
exact ⟨edit_lower_from_deficit (Nat.cast_nonneg P) hdef htarget.1 hsource.2,
encode_editDistance_le_real c x y⟩
/-- A code satisfying the additive block-pair estimate, with short-pair error
at most half a block, has both advertised inequalities and the stronger
one-sided-deficit inequality. The zero-deficit case is included. -/
theorem encode_bounds_of_additive {α : Type u} {β : Type v}
{M P error : ℕ} (c : α → Word β P) {x y : List α}
(hx : x.length = M) (hy : y.length = M)
(hadd : P * deficit M x y ≤
201 * deficit (M * P) (encode c x) (encode c y) + error)
(herr : 2 * error ≤ P) :
((P : ℝ) / 804 * (ed x y : ℝ) ≤
(ed (encode c x) (encode c y) : ℝ) ∧
(ed (encode c x) (encode c y) : ℝ) ≤
(P : ℝ) * (ed x y : ℝ)) ∧
(P : ℝ) / 402 * (deficit M x y : ℝ) ≤
(deficit (M * P) (encode c x) (encode c y) : ℝ) := by
have hdef := real_deficit_of_natural (natural_deficit_from_additive hadd herr)
exact ⟨encode_bounds_of_deficit c hx hy hdef, hdef⟩
/-- The source's real-valued short-pair threshold gives its integer loss
budget without rounding changes. -/
theorem natural_short_pair_error {M : ℕ}
(h : 2 * (M : ℝ) * segmentThreshold M ≤ (M : ℝ) ^ 2 / 2) :
2 * (2 * M * segmentThreshold M) ≤ M ^ 2 := by
have h' : (2 : ℝ) * (2 * (M : ℝ) * segmentThreshold M) ≤ (M : ℝ) ^ 2 := by
linarith
exact_mod_cast h'
end TreeEdit.BinaryLower.Code
end OAI
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