Mathlib v4.30 source proposition Nat.coprime_add_self_right

Identifier: F:ml430-nat-coprime-add-self-right-c3ed0f45
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m.Coprime (n + m) ↔ m.Coprime n

Dependencies

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

Evidence

kernel-Nat.coprime_add_self_right

Kind: kernel-term
Status: checked

Supports: gcd m (n+m) = 1 iff gcd m n = 1

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. Route: `g1 := gcd m (n+m)` divides `m` and `n+m` (`gcd_dvd_left`/`_right`); `dvd_add_iff_right` (after `add_comm` to match its `m+n` argument order) cancels the shared `m` factor to give `g1 | n`, so `dvd_gcd` gives `g1 | gcd m n =: g2`. Conversely `g2` divides `m` and `n`, so `dvd_add` gives `g2 | (n+m)` directly (already the lemma's own argument order, no reordering needed), and `dvd_gcd` gives `g2 | g1`. `dvd_antisymm` on those two divisibilities gives `g1 = g2`, from which the `Iff` on `= 1` follows by substitution. `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.coprime_add_self_right

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 coprime-backlog",
  "source": "statement-only extraction of `Nat.coprime_add_self_right` from Mathlib v4.30.0 (see prior_art), independently proved here as `Nat.coprime_add_self_right` and admitted through `Kernel::add_declaration`",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.coprime_add_self_right`",
      "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"
    }
  ]
}