parity_oneBitDiff_of_post

formal source

 130
 131/-! ## Main theorem: posting ⇒ parity adjacency -/
 132
 133theorem parity_oneBitDiff_of_post {d : Nat} (L : LedgerState d) (k : Fin d) (side : Side) :
 134    OneBitDiff (parity d L) (parity d (post L k side)) := by
 135  -- reduce to the coord-atomic step lemma and reuse `coordAtomicStep_oneBitDiff`
 136  have hstep := phiVec_coordAtomicStep_of_post (d := d) L k side
 137  simpa [parity] using (coordAtomicStep_oneBitDiff (d := d) (x := phiVec (d := d) L)
 138    (y := phiVec (d := d) (post L k side)) hstep)
 139
 140/-! ## Posting-step relation (ledger constraint ⇒ adjacency) -/
 141
 142/-- One atomic posting step between ledger states. -/
 143def PostingStep {d : Nat} (L L' : LedgerState d) : Prop :=
 144  ∃ k : Fin d, ∃ side : Side, L' = post L k side
 145
 146theorem postingStep_oneBitDiff {d : Nat} {L L' : LedgerState d} (h : PostingStep (d := d) L L') :
 147    OneBitDiff (parity d L) (parity d L') := by
 148  rcases h with ⟨k, side, rfl⟩
 149  simpa using parity_oneBitDiff_of_post (d := d) L k side
 150
 151/-! ## Optional deepening: a cost/legality predicate that implies `PostingStep` -/
 152
 153/-- L1 cost of a ledger transition, measured as total absolute change in debit+credit counts. -/
 154noncomputable def ledgerL1Cost {d : Nat} (L L' : LedgerState d) : Nat :=
 155  (∑ i : Fin d, Int.natAbs (L'.debit i - L.debit i)) +
 156  (∑ i : Fin d, Int.natAbs (L'.credit i - L.credit i))
 157
 158/-- Monotone-posting constraint: debit/credit counts never decrease. -/
 159def MonotoneLedger {d : Nat} (L L' : LedgerState d) : Prop :=
 160  (∀ i : Fin d, L.debit i ≤ L'.debit i) ∧ (∀ i : Fin d, L.credit i ≤ L'.credit i)
 161
 162/-- A small “legality” predicate: monotone ledger counts + unit L1 step. -/
 163def LegalAtomicTick {d : Nat} (L L' : LedgerState d) : Prop :=