feat(primality): add the HexPrimalityMathlib correspondence layer - #9412
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…visor lemmas powModNat gains the p = 0 guard so it agrees with powMod at every input, its recursion (powModNatGo, bitLength) is @[expose]d, its correctness theorem is exported, and an unconditional @[csimp] names powModNat the kernel-facing specification with powMod the runtime twin. instDecidablePrime re-routes decide through isPrimeTrial, exists_trial_divisor is exported, and exists_prime_dvd plus exists_prime_le_sqrt are the composite-witness lemmas the hex-primality Pocklington argument finishes with. SPEC amendments record the landed state (hex-arith SPEC modular-exponentiation and primality sections; hex-primality SPEC stale claims corrected). Progress: progress/20260822T034253Z_primality-m0.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
… at 10^4 Adds the HexPrimality library (deps HexArith, HexBasic) with its umbrella, libraries.yml entry, CI lib target, and the SPEC co-located at HexPrimality/SPEC/hex-primality.md. Table.lean commits the 1229-entry primeTable below primeTableBound = 10000 with sortedness, binary-search membership (isTablePrime_iff), soundness via one kernel-replayed isPrimeTrial sweep, completeness via five balanced kernel-replayed coverOk chunks, and primesIn with its membership theorem. Statements are sieve-agnostic so the planned kernel-reducible sieve swaps proof internals only. Progress: progress/20260822T040440Z_primality-table.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Order.lean defines orderOf as a bounded least-search with junk value 0 off the coprime domain and proves the SPEC's order stack Mathlib-free: coprimality from a unit power, coprime cancellation, multiplicative Fermat, order positivity via pigeonhole through the ListShim, order divisibility both ways, orderOf_dvd_pred, prime-power extraction into the order, and the two square roots of 1 modulo a prime. Progress: progress/20260822T041921Z_primality-order.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
…tion MillerRabin.lean adds oddSplit with its spec, the squaring loop, and millerRabin with the SPEC's exact branch table, plus defaultBases and isProbablePrime. Soundness is the contrapositive millerRabin_eq_true_of_prime by forward induction along the loop with sq_roots_of_one; not_prime_of_millerRabin_false is the public theorem, and no completeness is claimed. Guards cover Carmichael numbers, strong pseudoprimes, and agreement with isPrimeTrial below 512. Progress: progress/20260822T042306Z_primality-millerrabin.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Implements the hex-finite-field SPEC's Randomness section at its specified site in hex-basic: Rand with next, RandError carrying the advanced state on exhaustion, Rand.nat by rejection sampling that rejects the incomplete top interval, Rand.ofSeed, and the range theorem Rand.nat_lt. Known-answer guards pin the canonical splitmix64 seed-0 vector and deterministic replay. Progress: progress/20260822T042502Z_hexbasic-rand.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
…e checker Cert.lean defines PrimeCert (with the pock3 constructor carrying the integer-square-root witness the SPEC amendment records, its arithmetic stubbed false until milestone 4), the structurally recursive checkPrime/checkChildren pair with all workers @[expose]d against the kernel-facing powModNat, and checker soundness prime_of_checkPrime by strong induction on the subject value, through the pocklington core theorem, the fused coprime-prime-power combination, and the gcd-to-noncongruence transport. CheckedPrimeCert and the single-Bool-slot prime_of_checkPrimeAt serve the coming search and tactic layers. Guards cover accepted multi-level certificates and one rejected certificate of each kind. Progress: progress/20260822T043327Z_primality-cert.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Search.lean adds RhoStop/RhoFailure with the resumable-failure convention, rhoFactor? as Brent cycle search with restart draws from Hex.Rand and dynamic validation at every success exit (making rhoFactor?_spec a fuel induction over guards), and the internal partialFactor: table trial division, then a rho worklist whose every path preserves the product exactly, giving the unconditional partialFactor_prod invariant. Progress: progress/20260822T044003Z_primality-rho.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Search.lean gains the resumable failure types, the mutual fuel-and-worklist certificate search primeCertGo/assembleGo (runtime only, so well-founded recursion is fine), the indexed primeCert? whose runtime dependent checks build CheckedPrimeCert with zero proof plumbing, the verdict theorem primeCert?_composite by path analysis, isPrime? across the table/trial/certificate tiers with its exactness spec, the total isPrime with unconditional isPrime_iff, and nextPrime? with its leastness spec. Progress: progress/20260822T044458Z_primality-decide.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Elab.lean adds primality n and the three tactic forms per the SPEC: compiled certificate search at elaboration time from the reproducible seed, defeq transparency and bit-budget checks, verdict and exhaustion diagnostics, an untrusted-search self-check with the same compiled checkPrime the kernel replays, and an emitted term applying prime_of_checkPrimeAt to the reified certificate with one Eq.refl true slot. In-module tests cover every syntax form across all three decision tiers and the failure messages. Progress: progress/20260822T044810Z_primality-tactic.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
…rion Cert3.lean proves pocklington3, the BLS m = 1 theorem, Mathlib-free: divisor_mod_one lifts the per-prime-divisor congruence to all divisors; the certificate equation abF + (a+b) = 2sF + r pins the composite split; parity (R odd, F even) forces the mod-F offset even, which r < 2F kills on one side and the size-bound cap 2s <= 2F + r (two_s_le) kills on the other; the discriminant identity r^2 = (a-b)^2 + 8s then contradicts each disjunct of the condition. not_square_of_sqrt_witness is the two-multiplication witness form of the integer-square test for the checker. Progress: progress/20260822T045935Z_primality-bls.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
checkPock3Arith gains the real Brillhart-Lehmer-Selfridge conditions with the stored square-root witness; pock_divisor_step is factored out and shared by both soundness cases; the pock3 case of checker soundness closes through pocklington3; certificate search routes to mkPock3 when the factored part misses the square root, with the wrapper's checkPrime validation deciding acceptance. Guards cover the 199/F=6 cube-root acceptance and one rejection per condition. The SPEC records the except-step-4 correction, the witness-based discriminant test, and Nat.sqrt's removal from the replay closure. Progress: progress/20260822T050441Z_primality-pock3.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Scaffolds HexPrimalityMathlib and adds Prime.lean (prime_iff via prime_iff_forall_lt against Nat.prime_def_lt, plus Nat.Prime-flavoured transports of the checker, decision, Miller-Rabin, and next-prime theorems; no DecidablePred Nat.Prime instance, per SPEC) and Segment.lean (primeTable_spec, primesIn_spec, filter_prime_range, and the forall_prime_lt scaffold for segment statements). Progress: progress/20260822T050700Z_primality-companion.md Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
The stack changes factor-sweep-relevant sources with no effect on the factorization service: the powModNat kernel-facing amendments and the Decidable (Prime p) reroute in hex-arith, the primality conformance emitters, the hotPathCandidates proof-carrying view with pinned values, and per-milestone lakefile registrations. Each transition gets an exact blob-pair exemption with its reason, and the file is unioned with main's current entries so the stack also inherits the HexBasic/Rand attestation that merged separately. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
# Conflicts: # scripts/bench/proof_only_runtime_exemptions.json
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
# Conflicts: # scripts/bench/proof_only_runtime_exemptions.json
# Conflicts: # .github/workflows/ci.yml # SPEC/future-work.md
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
Codex review noted n.log2 alone admits an input one bit past the advertised ceiling; a positive numeral has n.log2 + 1 bits. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_0133rJ7x9mYiiXpJceCnc1A9
# Conflicts: # .github/workflows/ci.yml
# Conflicts: # HexPrimality.lean # scripts/bench/proof_only_runtime_exemptions.json
# Conflicts: # scripts/bench/proof_only_runtime_exemptions.json
# Conflicts: # HexPrimality.lean
# Conflicts: # .github/workflows/ci.yml # scripts/bench/proof_only_runtime_exemptions.json
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This PR scaffolds
HexPrimalityMathlib(milestone 5a of HexPrimality/SPEC/hex-primality.md) and lands the correspondence:prime_iffis the one lemma the whole layer turns on (prime_iff_forall_ltagainstNat.prime_def_lt), and everything else transports along it:natPrime_of_checkPrime,natPrime_of_checkPrimeAt(the statement the comingNat.Primetactic handler emits),isPrime_iff_natPrime,not_natPrime_of_millerRabin_false, andnextPrime?_natPrime. Per the SPEC, noDecidablePred Nat.Primeinstance is declared — Mathlib'sNat.decidablePrimealready exists and a second global instance would risk selection churn.Segment.leancarries the table statements in Mathlib's vocabulary:primeTable_spec,primesIn_spec,filter_prime_rangeoverFinset.range, andforall_prime_lt, the scaffold that turns "every prime belowxsatisfiesP" into a check over the committed table entries. The SPEC'slibraries.ymlblock is updated to the landed statuses.Stacked on #9411.
🤖 Prepared with Claude Code