File size: 3,483 Bytes
5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorCoverage
namespace OAI
/-! # Applying the four-block checker to original indexed top masks -/
namespace LowSpaceSubsetSum
structure TopCoordinates (n g : ℕ) where
index : Fin 4 × Fin g → Fin n
injective : Function.Injective index
namespace TopCoordinates
variable {n g : ℕ} (C : TopCoordinates n g)
def domain : Finset (Fin n) := Finset.univ.image C.index
def encodeMask (S : Finset (Fin n)) : Finset (Fin 4 × Fin g) :=
Finset.univ.filter (fun j => C.index j ∈ S)
@[simp] theorem encodeMask_mem (S : Finset (Fin n)) (j : Fin 4 × Fin g) :
j ∈ C.encodeMask S ↔ C.index j ∈ S := by simp [encodeMask]
def encodeEntry (x : Entry (Fin n)) : SeparatorCheck.Record g :=
⟨x.value, C.encodeMask x.mask⟩
theorem disjoint_of_encoded {S T : Finset (Fin n)} (hS : S ⊆ C.domain)
(h : Disjoint (C.encodeMask S) (C.encodeMask T)) : Disjoint S T := by
apply Finset.disjoint_left.mpr
intro i hi hj
obtain ⟨j, _, he⟩ := Finset.mem_image.mp (hS hi)
apply Finset.disjoint_left.mp h (C.encodeMask_mem S j |>.mpr (he ▸ hi))
exact (C.encodeMask_mem T j).mpr (he ▸ hj)
theorem encoded_disjoint {S T : Finset (Fin n)} (h : Disjoint S T) :
Disjoint (C.encodeMask S) (C.encodeMask T) := by
apply Finset.disjoint_left.mpr
intro j hj hk
exact Finset.disjoint_left.mp h ((C.encodeMask_mem S j).mp hj) ((C.encodeMask_mem T j).mp hk)
def check (families : Fin 4 → Array (Finset (Fin g))) (target : ℤ)
(xs ys : List (Entry (Fin n))) : SeparatorTraversal.Result :=
SeparatorCheck.run n families target (xs.map C.encodeEntry) (ys.map C.encodeEntry)
theorem sound (families : Fin 4 → Array (Finset (Fin g))) (target : ℤ)
(xs ys : List (Entry (Fin n))) (hsize : xs.length + ys.length ≤ 2 ^ n)
(htop : ∀ x ∈ xs, x.mask ⊆ C.domain)
(hyes : (C.check families target xs ys).accepted = true) :
∃ x ∈ xs, ∃ y ∈ ys, Disjoint x.mask y.mask ∧ x.value + y.value = target := by
obtain ⟨x, hxmap, y, hymap, hd, hw⟩ := SeparatorCheck.sound n families target
(xs.map C.encodeEntry) (ys.map C.encodeEntry) (by simpa only [List.length_map] using hsize) hyes
obtain ⟨x, hxOrig, rfl⟩ := List.mem_map.mp hxmap
obtain ⟨y, hyOrig, rfl⟩ := List.mem_map.mp hymap
exact ⟨x, hxOrig, y, hyOrig, C.disjoint_of_encoded (htop x hxOrig) hd, hw⟩
/-- The stored key witnesses suffice for original-input one-sided soundness. -/
theorem subset_sum_sound (I : Instance n) (F : Finset (Fin n))
(families : Fin 4 → Array (Finset (Fin g)))
(xs ys : List (Entry (Fin n))) (hsize : xs.length + ys.length ≤ 2 ^ n)
(leftDomain rightDomain : Finset (Fin n))
(hshared : leftDomain ∩ rightDomain ⊆ C.domain)
(hleft : ∀ x ∈ xs, Represents (flip (fun i => (I.values i : ℤ)) F) leftDomain C.domain x)
(hright : ∀ y ∈ ys, Represents (flip (fun i => (I.values i : ℤ)) F) rightDomain C.domain y)
(hyes : (C.check families ((I.target : ℤ) - weight (fun i => (I.values i : ℤ)) F)
xs ys).accepted = true) : Solvable I := by
have htop (x : Entry (Fin n)) (hx : x ∈ xs) : x.mask ⊆ C.domain := by
obtain ⟨S, _, _, hm⟩ := hleft x hx
rw [← hm]
exact Finset.inter_subset_right
obtain ⟨x, hx, y, hy, hd, hw⟩ := C.sound families _ xs ys hsize htop hyes
exact entry_match_sound I F hshared (hleft x hx) (hright y hy) hd hw
end TopCoordinates
end LowSpaceSubsetSum
end OAI
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