Mathlib v4.30 source proposition Nat.ModEq.cancel_left_div_gcd

Identifier: F:ml430-nat-modeq-cancel-left-div-gcd-57ef8287
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m a b c : ℕ}, 0 < m → c * a ≡ c * b [MOD m] → a ≡ b [MOD m / m.gcd c]

Dependencies

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9 direct dependencies 2 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.mod_eq_cancel_left_div_gcd

Kind: kernel-term
Status: checked

Supports: ∀ {m a b c : ℕ}, 0 < m → c * a ≡ c * b [MOD m] → a ≡ b [MOD m / m.gcd c]

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.mod_eq_cancel_left_div_gcd` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type. New proof, lane modeq-div-gcd, in `nat_prelude/modeq_cancel_div_gcd.rs`, wired in via `declare_modeq_cancel_div_gcd`. `Nat.gcd_mul_right` (landed for a sibling family within the hour before this lane started) turned out NOT to be what unlocks this family -- `Nat.gcd_cofactors_coprime` (`bezout.rs`, pre-existing) does: with `g := gcd(m,c)`, `Nat.div_mul_cancel_of_dvd` gives `g*(m/g)=m` and `g*(c/g)=c`, substituting those into `gcd c m = gcd m c = g` (`gcd_comm`) gives `gcd (g*(c/g)) (g*(m/g)) = g`, and `gcd_cofactors_coprime` turns that into `gcd (c/g) (m/g) = 1` directly. A local helper (`mod_eq_cancel_scale`) then cancels the shared factor `g` from both the modulus and both endpoints of the rewritten hypothesis (`left_distrib`/`mul_assoc`/`Nat.mul_left_cancel_of_pos`, peeling `ModEq`'s nested existential the same way `euler.rs`'s `cancel_common_right_addend` does, with the SAME witnesses surviving), and the pre-existing coprime `Nat.mod_eq_cancel` finishes. `nat_theorem_inventory`'s rendered type matches this fact's `formal.statement` exactly (mod carrier-name rewriting: `AxNat` for `ℕ`, `AxNat.modEq`/`[MOD]`, `AxNat.lt`/`<`, `AxNat.div`/`/`, `AxNat.gcd`/`.gcd`). `nat_theorem_inventory` exits non-zero (0 rows) for a bogus name (`mod_eq_cancel_left_div_gcd_bogus`, measured). Verified no substring overlap against the sibling fact `Nat.mod_eq_cancel_left_div_gcd_general` (this fact's third mirror, `F:ml430-nat-modeq-cancel-left-div-gcd-cfca1225`): the anchored pattern `^Nat\.mod_eq_cancel_left_div_gcd[[:space:]]` matches exactly 1 row (its own), not the `_general` variant, in the combined inventory of all three new theorems.

footprint-Nat.mod_eq_cancel_left_div_gcd

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 (measured: axiom=0 opaque=0 quotient=0). `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly, and `mod_eq_cancel_div_gcd_family_applies_at_a_discriminating_concrete_instance_and_symbolically` checks it applies at a concrete DISCRIMINATING instance (m,c)=(6,4), gcd=2>1 -- a coprime instance could not tell this family apart from the pre-existing `Nat.mod_eq_cancel` -- with a transposed negative control, and symbolically at a genuinely free (m,a,b,c) via a fresh restating theorem.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.ModEq.cancel_left_div_gcd` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.ModEq.cancel_left_div_gcd`",
      "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"
    }
  ]
}