Mathlib v4.30 source proposition Nat.ModEq.of_mul_left

Identifier: F:ml430-nat-modeq-of-mul-left-88d20bca
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.ModEq.of_mul_left` in the pinned Mathlib v4.30 source.

Formal statement

∀ {n a b : ℕ} (m : ℕ), a ≡ b [MOD m * n] → a ≡ b [MOD n]

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

0 direct dependencies 1 direct dependents

Evidence

conclusion-directed-ofMulLeft

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Nat.ModEq.of_mul_left` in the pinned Mathlib v4.30 source, admitted here by `Kernel::add_declaration` over a proof term the conclusion-directed producer built and the same kernel re-checked: axiom footprint empty, `target_dependency` false, and exactly one cited theorem dependency, `Axeyum.Autogenesis.Candidate.NatModEqCongruence.ofMulLeft`, whose own Lean proof is authored in the tracked contract source and is itself axiom-free.

Checker command

test "$(python3 scripts/check-autogenesis-nat-modeq-congruence-family.py 2>&1 | grep -Ec '^NAT_MODEQ_CONGRUENCE\|PASS$')" -ge 1

Evidence notes

The checker_command's exit status depends on what the run FOUND: the script replays all ten targets against the pinned external exports and fails on any changed digest, a nonempty axiom footprint, a new theorem dependency, a target self-citation, a ledger row that disagrees, and on the probe accepting a nonexistent input. `[[:space:]]`-free by construction — the grep matches a whole literal line. Per ADR-0601 the cited candidate is a producer behind the one trust anchor, not a Mathlib proof import: no Mathlib proof value for this theorem was exposed.

Provenance

{
  "date": "2026-08-18",
  "established_by": "lane modeq-producer (2026-08-28): the conclusion-directed transport of one axiom-free Lean candidate contract (scripts/lean/autogenesis_nat_modeq_congruence_contract_v1.lean) into a proof-isolated import of this statement. Every public Lean 4.30 `Nat` remainder lemma carries `propext` (measured: Nat.mod_zero, Nat.mod_eq_of_lt, Nat.add_mod, Nat.mod_mod_of_dvd, Nat.mod_self), so the contract rebuilds the `Nat.mod` recurrence over Nat.modCore/Nat.modCore.go and derives every congruence law from it by structural fuel induction.",
  "prior_art": [
    {
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted",
      "what": "the theorem declaration `Nat.ModEq.of_mul_left`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "who": "the Mathlib contributors",
      "year": 2026
    }
  ],
  "source": "statement-only extraction of `Nat.ModEq.of_mul_left` from Mathlib v4.30.0; no proof value was exposed"
}