Mathlib v4.30 source proposition Nat.mod_mul_left_div_self

Identifier: F:ml430-nat-mod-mul-left-div-self-0aca6c6e
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (m n k : ℕ), m % (k * n) / n = m / n % k

Dependencies

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

Evidence

kernel-Nat.mod_mul_left_div_self

Kind: kernel-term
Status: checked

Supports: ∀ (m n k : ℕ), m % (k * n) / n = m / n % k

Checker command

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

Evidence notes

Declared in `nat_prelude/mod_mul_lemmas.rs`'s `declare_mod_mul_family`. Case-splits `n` (outer `cases_zero_succ`): at `n=0` both sides collapse to `zero` via `mul_zero`/`mod_zero`/`div_zero`/`zero_mod` congruence (needs no positivity at all); at `n=succ npred`, case-splits `k`: at `k=0` both sides collapse to `div m n` via `zero_mul`/`mod_zero`; at `k=succ kpred` (both positive), the local helper `mod_mul_div_self` answers `div (mod m e) n = mod (div m n) k` for `e = mul n k` (bridged from this fact's `k*n` via `mul_comm`) -- chains `mod_mul`'s identity (`F:ml430-nat-mod-mul-beaccbad`) to rewrite `m%(n*k)` as `m%n + n*(m/n%k)`, then `add_mul_div_left` to divide by `n`, landing on `(m%n)/n + (m/n%k)`; the local helper `div_of_lt` (manufacture a second trivial `divMod` and compare via `div_mod_unique`) collapses `(m%n)/n` to `0` via `mod_lt`, and `zero_add` finishes. `nat_theorem_inventory`'s rendered type matches this fact's `formal.statement` verbatim (`x0`=m, `x1`=n, `x2`=k).

footprint-Nat.mod_mul_left_div_self

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). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds Nat.mod_mul_left_div_self's own footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly (via `theorem_names`, which now lists it).

Provenance

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