Mathlib v4.30 source proposition Nat.mod_lcm

Identifier: F:ml430-nat-mod-lcm-ee6bdd41
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n a b : ℕ}, a ≡ b [MOD n] → a ≡ b [MOD m] → a ≡ b [MOD n.lcm m]

Dependencies

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14 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.mod_lcm

Kind: kernel-term
Status: checked

Supports: a ≡ b [MOD n] and a ≡ b [MOD m] together imply a ≡ b [MOD lcm n m], unconditionally in n and m

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. Unlike `Nat.crt_unique` (this prelude's other congruence-combination theorem), no `gcd n m = 1` hypothesis is needed: the divisibility combination step is `Nat.lcm_dvd : dvd n c -> dvd m c -> dvd (lcm n m) c`, which is already unconditional (unlike `coprime_mul_dvd`, which needs coprimality to rewrite `lcm n m` down to `n*m`). Route: `le_total x y` splits into the two orders; each order extracts `dvd n (sub y x)` and `dvd m (sub y x)` from the two congruences (the same `gap_dvd` step `crt_unique` uses), combines them with `lcm_dvd`, and repackages the result as `modEq (lcm n m) x y` (`crt_unique`'s `modeq_of_dvd_gap`); the `y <= x` branch flips both hypotheses and the conclusion through `mod_eq_symm`. `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.mod_lcm

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 [].

compute-Nat.mod_lcm

Kind: kernel-term
Status: checked

Supports: the theorem genuinely combines two non-coprime moduli via their lcm, not merely their product

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::mod_lcm_holds_at_a_concrete_instance_without_coprimality

Evidence notes

Applies the theorem at n=4, m=6 (gcd 4 6 = 2, so `crt_unique` could not prove this instance) with x=1, y=13: 1 congruent to 13 mod 4 and mod 6, and lcm 4 6 computes to 12, so the theorem must admit modEq 12 1 13. NEGATIVE CONTROL: the identical proof term reused against modulus 5 (not lcm 4 6) is rejected by the kernel with a TypeMismatch/DeclarationValueMismatch.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-singles",
  "source": "statement-only extraction of `Nat.mod_lcm` from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.mod_lcm` (`crates/axeyum-lean-kernel/src/nat_prelude/lcm.rs`) and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.mod_lcm`",
      "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, and this project's proof is independently constructed"
    }
  ]
}