Mathlib v4.30 source proposition Nat.Coprime.coprime_div_left

Identifier: F:ml430-nat-coprime-coprime-div-left-6f7082bd
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n a : ℕ}, m.Coprime n → a ∣ m → (m / a).Coprime n

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

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

Evidence

kernel-Nat.coprime_div_left

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Nat.Coprime.coprime_div_left` in the pinned Mathlib v4.30 source.

Checker command

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

Evidence notes

`Nat.coprime_div_left` in `crates/axeyum-lean-kernel/src/nat_prelude/coprime_lemmas.rs (declare_coprime_div_left)`. Mirror image of `Nat.coprime_div_right`: cases on the divisor `a`, dividing `m` instead of `n`. At `a = 0`, `dvd 0 m` forces `m = 0` (`zero_mul`) and `div _ 0 = 0` (`div_zero`) collapse both sides; at `a = succ a'`, the witness from `dvd a m` recovers `div m a` via `div_mul_cancel_of_dvd`, and the succ-branch shrink step uses `coprime_of_dvd_left` (shrinking the LEFT `gcd` argument) instead of `coprime_of_dvd_right`. Pinned by a concrete-instantiation test exercising both branches (`nat_prelude_tests::coprime_div_left_applies_at_both_branches_of_its_case_split`).

footprint-Nat.coprime_div_left

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- this theorem's trusted closure is empty

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.coprime_div_left 2>/dev/null | /usr/bin/grep -xFc $'nat\tNat.coprime_div_left\t0\t')" -ge 1

Evidence notes

The matched row is `nat<TAB>Nat.coprime_div_left<TAB>0<TAB>` with an EMPTY trailing footprint column, so a nonzero footprint or an absent declaration both give a count of 0 and fail. Uses `/usr/bin/grep` explicitly (not the interactive `ugrep` alias) and ANSI-C-quoted literal tabs, per this repository's own grep-dialect gotcha. Run `--release`: in debug, building the full constructed environment stack-overflows, which is a resource limit and not a fact about this theorem.

Provenance

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