File size: 3,443 Bytes
ea76aac | 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 | import Mathlib
import OAI.Computability.MaxCut.Machines.MachinePaddedExpanderFamilyBounds
namespace OAI
/-!
# Exact serialized family tables at actual regularization vertices
The geometric family maps request zero to a one-vertex table. Empty clouds in
regularization instead remain empty. The bridge therefore requires positive
cloud size, derived here from each actual original or dummy vertex.
-/
namespace MaxCutGames.Foundations.PCP.PreprocessingFamilyBridge
open PreprocessingCloudIndex PreprocessingRegularTables ExpanderTableWords
open MaxCutGames.Foundations.Complexity (encodeWords)
/-- Transporting only the size equality leaves every serialized rotor entry
unchanged; no relabeling or alternative numbering is introduced. -/
theorem rotationWords_resizeTable {n m q : Nat} (h : n = m)
(table : ExpanderTables.Table n q) :
rotationWords (resizeTable h table) = rotationWords table := by
cases h
rfl
/-- This left side is definitionally the output `MachineRegularFamily.rotor`
for the request on its physical core tape 4. The right side is exactly the
archive expected on core tape 7 by the internal-row machine. -/
theorem familyRotor_eq_familyCloudTable (H : BaseTable) (t : GraphTables.Table)
(v : Fin t.vertices) (hk : 0 < cloudSize t v) :
encodeWords (rotationWords (ExpanderTables.family H
(PreprocessingLevels.boundedLevel (cloudSize t v)))) =
encodeWords (rotationWords (familyCloudTable H t v)) := by
unfold familyCloudTable
rw [dite_eq_right (Nat.ne_of_gt hk)]
erw [rotationWords_resizeTable]
rfl
/-- An actual original dart supplies a member of its owner's cloud. -/
theorem cloudSize_pos_of_dart (t : GraphTables.Table) (e : Fin t.darts) :
0 < cloudSize t t.rows[e].tail := by
have h := (oldCloudIndex t e).isLt
omega
/-- An empty cloud has no padding vertices either. -/
theorem padding_zero_of_cloudSize_zero (t : GraphTables.Table) (v : Fin t.vertices)
(hk : cloudSize t v = 0) : padding t v = 0 := by
simp only [padding, hk, PreprocessingLevels.cloudPaddedSize_zero, Nat.sub_zero]
/-- Every actual dummy vertex also belongs to a positive original cloud. -/
theorem cloudSize_pos_of_dummy (t : GraphTables.Table) (v : Fin t.vertices)
(j : Fin (padding t v)) : 0 < cloudSize t v := by
by_contra h
have hk := Nat.eq_zero_of_not_pos h
have hp := padding_zero_of_cloudSize_zero t v hk
have hj := j.isLt
omega
/-- The same positivity fact for the common local coordinate interface. -/
theorem cloudSize_pos_of_paddedCloud (t : GraphTables.Table) (v : Fin t.vertices)
(x : PaddedCloud t (padding t) v) : 0 < cloudSize t v := by
by_contra h
have hk := Nat.eq_zero_of_not_pos h
have hp := padding_zero_of_cloudSize_zero t v hk
have hx := (paddedCloudRank t (padding t) v x).isLt
omega
/-- The zero-cloud branch remains genuinely empty, including its serialized
rotor archive. It is not replaced by the geometric family at request zero. -/
theorem familyCloudWords_of_zero (H : BaseTable) (t : GraphTables.Table)
(v : Fin t.vertices) (hk : cloudSize t v = 0) :
encodeWords (rotationWords (familyCloudTable H t v)) = [] := by
have hl : (rotationWords (familyCloudTable H t v)).length = 0 := by
rw [rotationWords_length, cloudSize_add_padding, hk,
PreprocessingLevels.cloudPaddedSize_zero, Nat.zero_mul]
rw [List.length_eq_zero_iff.mp hl]
rfl
end MaxCutGames.Foundations.PCP.PreprocessingFamilyBridge
end OAI
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