Mathlib v4.30 source proposition Nat.ModEq.cancel_left_div_gcd'

Identifier: F:ml430-nat-modeq-cancel-left-div-gcd-cfca1225
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 d : ℕ}, 0 < m → c ≡ d [MOD m] → c * a ≡ d * b [MOD m] → a ≡ b [MOD m / m.gcd c]

Dependencies

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4 direct dependencies 0 direct dependents

Evidence

kernel-Nat.mod_eq_cancel_left_div_gcd_general

Kind: kernel-term
Status: checked

Supports: ∀ {m a b c d : ℕ}, 0 < m → c ≡ d [MOD m] → c * a ≡ d * 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_general 2>/dev/null | grep -Ec '^Nat\.mod_eq_cancel_left_div_gcd_general[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.mod_eq_cancel_left_div_gcd_general` (Mathlib's `Nat.ModEq.cancel_left_div_gcd'` -- Rust identifiers cannot carry `'`, so the field/kernel name carries the `_general` suffix this development uses for other primed Mathlib names) through the trusted `Kernel::add_declaration` gate. New proof, lane modeq-div-gcd, in `nat_prelude/modeq_cancel_div_gcd.rs`. Not re-derived from scratch: reduces to `Nat.mod_eq_cancel_left_div_gcd` via `Nat.mod_eq_mul_right`/`Nat.mod_eq_trans` (`c ≡ d [MOD m]` symmetrized and scaled by `b` gives `d*b ≡ c*b [MOD m]`, transitively chained against the hypothesis `c*a ≡ d*b [MOD m]` to reach `c*a ≡ c*b [MOD m]`), exactly Mathlib's own proof (`(h.trans (hcd.symm.mul_right b)).cancel_left_div_gcd hm`). `nat_theorem_inventory`'s rendered type matches this fact's `formal.statement` exactly. `nat_theorem_inventory` exits non-zero (0 rows) for a bogus name (measured), and does not collide with the sibling `F:ml430-nat-modeq-cancel-left-div-gcd-57ef8287` under the anchored pattern (see that fact's own evidence note).

footprint-Nat.mod_eq_cancel_left_div_gcd_general

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"
    }
  ]
}