Mathlib v4.30 source proposition Nat.dvd_lcm_of_dvd_left

Identifier: F:ml430-nat-dvd-lcm-of-dvd-left-141a64bb
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b : ℕ}, a ∣ b → ∀ (c : ℕ), a ∣ b.lcm c

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dvd_lcm_of_dvd_left

Kind: kernel-term
Status: checked

Supports: a divides b implies, for every c, a divides lcm b c

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- dvd_lcm_of_dvd_left 2>/dev/null | grep -Ec '^Nat\.dvd_lcm_of_dvd_left[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the constructed proof term against the stated type: `dvd_trans` composing the hypothesis `a | b` with `dvd_lcm_left : b | lcm b c` (declared in `nat_prelude/lcm.rs`, read here through the shared `NatPrelude` fields, not edited). `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` requires the admitted declaration to be printed.

footprint-Nat.dvd_lcm_of_dvd_left

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration. A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-gcd",
  "source": "statement-only extraction of the Mathlib proposition from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.dvd_lcm_of_dvd_left` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dvd_lcm_of_dvd_left`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}