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e5a97c6 | 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 | import OAI.Combinatorics.MatrixRemoval.HostSampledSelection
import OAI.Combinatorics.MatrixRemoval.HostSamplingCompatibility
namespace OAI
/-!
# Arbitrary-repair lower bound for the actual canonical host
The concrete common-leaf selection satisfies both independent certificates:
its guarded path is valid, and its chosen coordinates bound each guard cell's
sampling multiplicity.
Together these prove the integer repair lower bound. No assumption
about preservation of arbitrary variable-variable cells is made.
-/
noncomputable section
namespace Problem348.Construction.SampledSelection
open HostSampledPath HostSampling TreeSampling
/-- The actual constant-offset anchor frame is chosen by its row/column seeds. -/
theorem frame_chosen {h n : ℕ} (er ec : Position h ≃ Fin n)
(ha : AxisConditions er ec) (s : Seeds (2 ^ h)) (t : Mode h) :
FrameChosen er ec s (frame er ec ha t s.2.1 s.2.2) := by
constructor
· intro u
simp [frame, SelectedAnchor.frame, sampleDepth]
· intro v
simp [frame, SelectedAnchor.frame, sampleDepth]
· intro u
exact rowChosen_path_anchor er s t u
· intro v
exact colChosen_path_anchor ec s t v
/-- Every actual path point and anchor frame obeys the uniform sampler's
coordinate equations. This is separate from all matrix-entry claims. -/
theorem selection_chosen {h n : ℕ} (hh : 1 ≤ h) (er ec : Position h ≃ Fin n)
(ha : AxisConditions er ec) (s : Seeds (2 ^ h)) :
PathChosen er ec s (selection hh er ec ha s.1 s.2.1 s.2.2) := by
constructor
· intro i hi
simpa only [selection, path_of_le hi] using rowChosen_path er i hi true s
· intro i hi
simpa only [selection, path_of_le hi] using rowChosen_path er i hi false s
· intro i hi
simpa only [selection, path_of_le hi] using colChosen_path ec i hi true s
· intro i hi
simpa only [selection, path_of_le hi] using colChosen_path ec i hi false s
· exact rowChosen_path_dummy er s
· exact colChosen_path_dummy ec s
· simp [selection, HostSampledPath.dummy, sampleDepth]
· simp [selection, HostSampledPath.dummy, sampleDepth]
· simpa only [selection, Equiv.symm_apply_apply, path_of_le (Nat.zero_le h)] using
protected_path_roots h true true s.1 s.2.1 s.2.2
· simpa only [selection, Equiv.symm_apply_apply, path_of_le (Nat.zero_le h)] using
protected_path_roots h false false s.1 s.2.1 s.2.2
· intro i hi
exact frame_chosen er ec ha s _
· intro i hi
exact frame_chosen er ec ha s _
· intro i hi
exact frame_chosen er ec ha s _
· intro i hi
exact frame_chosen er ec ha s _
/-- Each fixed guarded cell occupies at most the prescribed `1/m²` fraction
of the three-seed sample space, for the actual selected paths. -/
theorem selection_guard_frequency {h n : ℕ} (hh : 1 ≤ h)
(er ec : Position h ≃ Fin n) (ha : AxisConditions er ec)
(rc : Fin n × Fin n) :
(Finset.univ.filter fun s : Seeds (2 ^ h) =>
rc ∈ (selection hh er ec ha s.1 s.2.1 s.2.2).guards h).card * (2 ^ h) ^ 2 ≤
Fintype.card (Seeds (2 ^ h)) :=
path_guard_frequency er ec
(fun s => selection hh er ec ha s.1 s.2.1 s.2.2)
(fun s => selection_chosen hh er ec ha s) rc
/-- The arbitrary-repair lower bound for the literal canonical host.
The independent axis enumerations satisfy `AxisConditions`: increasing anchor
groups precede all nonanchors, variable rows precede dummy rows, dummy columns
precede variable columns, and sampled parent/child pairs obey the four signed
tree-order inequalities. No entry-preservation condition is imposed on repairs. -/
theorem hammingDistance_ge {h n : ℕ} (hh : 1 ≤ h)
(er ec : Position h ≃ Fin n) (ha : AxisConditions er ec)
(B : BinaryMatrix n) (hfree : HFree fixedH B) :
(2 ^ h) ^ 2 ≤ hammingDistance (SelectedAnchor.matrix er ec) B := by
classical
apply Sampling.hammingDistance_ge_of_guard_sampling fixedH
(SelectedAnchor.matrix er ec) B
(fun s : Seeds (2 ^ h) => (selection hh er ec ha s.1 s.2.1 s.2.2).guards h)
(2 ^ h)
· rw [card_seeds]
positivity
· exact selection_guard_frequency hh er ec ha
· intro s hag
exact GuardedPath.Selection.not_HFree_of_agree
(selection_valid hh er ec ha s.1 s.2.1 s.2.2) hag
· exact hfree
end Problem348.Construction.SampledSelection
end
end OAI
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