toLocalConfigSpace

formal source

  89
  90    Note: Full implementation requires showing R̂ neighborhoods satisfy
  91    the refinement property. This structural version shows the pattern. -/
  92noncomputable def toLocalConfigSpace {L : Type*} [RSConfigSpace L]
  93    (rs : RSLocalityFromRHat L) : LocalConfigSpace L where
  94  N := fun ℓ => {U | ∃ k : ℕ, k > 0 ∧ U = rs.RHat ℓ}  -- Single step for now
  95  -- For a full implementation, would use k-step R̂ neighborhoods
  96  mem_of_mem_N := fun ℓ U ⟨k, hk, hU⟩ => hU ▸ rs.self_in_RHat ℓ
  97  N_nonempty := fun ℓ => ⟨rs.RHat ℓ, 1, Nat.one_pos, rfl⟩
  98  intersection_closed := fun ℓ U V ⟨k₁, hk1, hU⟩ ⟨k₂, hk2, hV⟩ => by
  99    -- Both U and V are rs.RHat ℓ, so their intersection is rs.RHat ℓ
 100    subst hU hV
 101    refine ⟨rs.RHat ℓ, ⟨1, Nat.one_pos, rfl⟩, ?_⟩
 102    rw [Set.inter_self]
 103  refinement := fun ℓ U ℓ' ⟨k, hk, hU⟩ hℓ' => by
 104    subst hU
 105    -- We need: ∃ W ∈ N(ℓ'), W ⊆ RHat ℓ
 106    -- N(ℓ') = {W | ∃ k, k > 0 ∧ W = RHat ℓ'}, so W = RHat ℓ'
 107    -- Need: RHat ℓ' ⊆ RHat ℓ (recognition transitivity)
 108    refine ⟨rs.RHat ℓ', ⟨1, Nat.one_pos, rfl⟩, ?_⟩
 109    exact rs.transitivity ℓ ℓ' hℓ'
 110
 111/-! ## RS Recognizers from Measurement -/
 112
 113/-- **Structural Definition**: Measurement maps in RS are recognizers.
 114
 115    A measurement in RS:
 116    - Takes a ledger state ℓ
 117    - Returns an observable event e
 118    - Two states with the same measurement outcome are indistinguishable
 119
 120    The 8-tick cadence ensures measurements have finite local resolution. -/
 121structure RSMeasurement (L E : Type*) [RSConfigSpace L] where
 122  /-- The measurement function -/