prime_logic_is_positive_ledger_state
plain-language theorem explainer
If a logic natural p recovers to a classical prime under the toNat map, then p satisfies the positive ledger state predicate 0 < p. Number theorists building the logic-native prime ledger cite this to confirm recovered atoms remain strictly positive. The proof is a one-line term that invokes the order isomorphism toNat_lt on zero and p, then simplifies the positivity of primes via toNat_zero.
Claim. Let $p$ be an element of the logic naturals. If the recovered value of $p$ is a prime natural number, then $0 < p$.
background
LogicNat is the inductive type with constructors identity (the zero-cost element) and step, forming the smallest orbit closed under multiplication by the generator. The function toNat extracts the iteration count, sending identity to 0 and step n to the successor of toNat n. IntegerLedgerStateLogic n is the proposition 0 < n, the recovered positive integer ledger state.
proof idea
The term applies toNat_lt to zero and p, using the mpr direction to obtain zero < p from toNat zero < toNat p. The hypothesis that toNat p is prime supplies positivity in Nat, which is rewritten via toNat_zero and simplified to close the inequality.
why it matters
The result supplies the prime_positive field of the PrimeLedgerLogicCert certificate that ties recovered prime atoms to the classical prime ledger. It closes the positivity direction needed for the equivalence between logic-native primality and ledger-atom status in the RH adapter stack.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.