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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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.Combinatorics.BinaryEdit.EncodingUpper
import OAI.Combinatorics.BinaryEdit.EditMetric
import OAI.Combinatorics.BinaryEdit.Distortion
namespace OAI
/-! Distortion transport through finite images of fixed-width codes. -/
noncomputable section
namespace TreeEdit.BinaryLower
universe u
/-- The concatenation extension, viewed in its exact-length target type. -/
def encodedWord {α : Type u} {M P : ℕ} (c : α → Word Bool P)
(x : Word α M) : Word Bool (M * P) :=
⟨encode c x.val, by rw [length_encode, x.property]⟩
/-- The actual finite image, rather than an abstract copy of the source set. -/
def encodedSet {α : Type u} {M P : ℕ} (c : α → Word Bool P)
(W : Finset (Word α M)) : Finset (Word Bool (M * P)) :=
W.image (encodedWord c)
lemma encodedWord_injective_of_lower {α : Type u} {M P : ℕ}
(c : α → Word Bool P) {a : ℝ} (ha : 0 < a)
(hlower : ∀ x y : Word α M,
a * (ed x.val y.val : ℝ) ≤
(ed (encode c x.val) (encode c y.val) : ℝ)) :
Function.Injective (encodedWord (M := M) c) := by
intro x y hxy
apply Subtype.ext
apply ed_eq_zero_iff.mp
have hval : encode c x.val = encode c y.val := congrArg Subtype.val hxy
have h := hlower x y
rw [hval, ed_self, Nat.cast_zero] at h
have hed : (ed x.val y.val : ℝ) ≤ 0 := nonpos_of_mul_nonpos_right h ha
exact Nat.eq_zero_of_le_zero (by exact_mod_cast hed)
lemma encodedSet_card {α : Type u} {M P : ℕ}
(c : α → Word Bool P) {a : ℝ} (ha : 0 < a)
(hlower : ∀ x y : Word α M,
a * (ed x.val y.val : ℝ) ≤
(ed (encode c x.val) (encode c y.val) : ℝ))
(W : Finset (Word α M)) : (encodedSet c W).card = W.card := by
exact Finset.card_image_of_injective W (encodedWord_injective_of_lower c ha hlower)
/-- The map from source membership witnesses into membership witnesses of the
encoded finite set, with no choice of an inverse. -/
def encodedMember {α : Type u} {M P : ℕ} (c : α → Word Bool P)
(W : Finset (Word α M)) : {x : Word α M // x ∈ W} →
{y : Word Bool (M * P) // y ∈ encodedSet c W} :=
fun x => ⟨encodedWord c x.val, Finset.mem_image_of_mem _ x.property⟩
lemma encodedMember_injective {α : Type u} {M P : ℕ}
(c : α → Word Bool P) {a : ℝ} (ha : 0 < a)
(hlower : ∀ x y : Word α M,
a * (ed x.val y.val : ℝ) ≤
(ed (encode c x.val) (encode c y.val) : ℝ))
(W : Finset (Word α M)) : Function.Injective (encodedMember c W) := by
intro x y hxy
apply Subtype.ext
apply encodedWord_injective_of_lower c ha hlower
exact congrArg Subtype.val hxy
/-- Distortion transport for the concrete finite image of a bi-Lipschitz code. -/
lemma c1_le_encodedSet_mul {α : Type u} {M P : ℕ}
(c : α → Word Bool P) (W : Finset (Word α M)) (hW : 2 ≤ W.card)
{a b : ℝ} (ha : 0 < a) (hb : 0 < b)
(hlower : ∀ x y : Word α M,
a * (ed x.val y.val : ℝ) ≤
(ed (encode c x.val) (encode c y.val) : ℝ))
(hupper : ∀ x y : Word α M,
(ed (encode c x.val) (encode c y.val) : ℝ) ≤
b * (ed x.val y.val : ℝ)) :
c1 {x : Word α M // x ∈ W}
(fun x y => (ed x.val.val y.val.val : ℝ)) ≤
(b / a) * binarySetDistortion (encodedSet c W) := by
classical
have : Nontrivial {x : Word α M // x ∈ W} := by
obtain ⟨x, hx, y, hy, hxy⟩ := Finset.one_lt_card.mp (show 1 < W.card by omega)
refine ⟨⟨⟨x, hx⟩, ⟨y, hy⟩, ?_⟩⟩
exact fun h => hxy (congrArg Subtype.val h)
have hg := encodedMember_injective c ha hlower W
have : Nontrivial {y : Word Bool (M * P) // y ∈ encodedSet c W} := hg.nontrivial
unfold binarySetDistortion
refine c1_le_bilip_mul _ _ ?_ ?_ ?_ (encodedMember c W) hg a b ha hb ?_ ?_
· intro x y
exact Nat.cast_nonneg _
· intro x y
exact Nat.cast_nonneg _
· intro x y hxy
apply Nat.cast_pos.mpr
apply editDistance_pos_iff.mpr
intro h
exact hxy (Subtype.ext (Subtype.ext h))
· intro x y
exact hlower x.val y.val
· intro x y
exact hupper x.val y.val
/-- The scale cancels: lower factor `P/804` and upper factor `P` lose only `804`. -/
lemma c1_le_804_encodedSet {α : Type u} {M P : ℕ}
(c : α → Word Bool P) (hP : 0 < P)
(W : Finset (Word α M)) (hW : 2 ≤ W.card)
(hlower : ∀ x y : Word α M,
((P : ℝ) / 804) * (ed x.val y.val : ℝ) ≤
(ed (encode c x.val) (encode c y.val) : ℝ)) :
c1 {x : Word α M // x ∈ W}
(fun x y => (ed x.val.val y.val.val : ℝ)) ≤
804 * binarySetDistortion (encodedSet c W) := by
have hPR : (0 : ℝ) < P := Nat.cast_pos.mpr hP
have h := c1_le_encodedSet_mul c W hW (a := (P : ℝ) / 804) (b := (P : ℝ))
(by positivity) hPR hlower (fun x y => encode_editDistance_le_real c x.val y.val)
have hratio : (P : ℝ) / ((P : ℝ) / 804) = 804 := by
field_simp
simpa only [hratio] using h
/-- Existential form used by the finite-tree binary-family construction. -/
theorem exists_binary_image_of_code {α : Type u} {M P : ℕ}
(c : α → Word Bool P) (hP : 0 < P)
(W : Finset (Word α M)) (hW : 2 ≤ W.card)
(hlower : ∀ x y : Word α M,
((P : ℝ) / 804) * (ed x.val y.val : ℝ) ≤
(ed (encode c x.val) (encode c y.val) : ℝ)) :
∃ V : Finset (Word Bool (M * P)), 2 ≤ V.card ∧
c1 {x : Word α M // x ∈ W}
(fun x y => (ed x.val.val y.val.val : ℝ)) / 804 ≤
binarySetDistortion V := by
have hPR : (0 : ℝ) < P := Nat.cast_pos.mpr hP
refine ⟨encodedSet c W, ?_, ?_⟩
· rw [encodedSet_card c (by positivity : (0 : ℝ) < P / 804) hlower W]
exact hW
· apply (div_le_iff₀ (by norm_num : (0 : ℝ) < 804)).mpr
simpa only [mul_comm] using c1_le_804_encodedSet c hP W hW hlower
end TreeEdit.BinaryLower
end
end OAI
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