Mathlib v4.30 source proposition Nat.mod_mul

Identifier: F:ml430-nat-mod-mul-beaccbad
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b x : ℕ}, x % (a * b) = x % a + a * (x / a % b)

Dependencies

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

Evidence

kernel-Nat.mod_mul

Kind: kernel-term
Status: checked

Supports: ∀ {a b x : ℕ}, x % (a * b) = x % a + a * (x / a % b)

Checker command

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

Evidence notes

Declared in `nat_prelude/mod_mul_lemmas.rs`'s `declare_mod_mul_family`. Case-splits `a` (`cases_zero_succ`): at `a=0` the whole identity collapses via `zero_mul`/`mod_zero`/`add_zero` congruence, never evaluating `mod _ 0` beyond the one declared equation; at `a=succ apred`, case-splits `b`: at `b=0` the identity again collapses via `mul_zero`/`mod_zero` plus the canonical `x = a*(x/a)+x%a` decomposition (`div_mod_exec`) commuted; at `b=succ bpred` (both positive), `double_decompose` reconstructs `divMod (a*b) x ((x/a)/b) (x%a + a*(x/a%b))` directly from two `div_mod_exec` decompositions (`x` at `a`, then `x/a` at `b`) combined via `left_distrib`/`mul_assoc`/`add_assoc`/`add_comm`, and `div_mod_unique` against the canonical decomposition of `x` at `a*b` (positivity from `one_le_mul`) forces the remainder equality directly. `nat_theorem_inventory`'s rendered type for `Nat.mod_mul` matches this fact's `formal.statement` verbatim (`x0`=a, `x1`=b, `x2`=x).

footprint-Nat.mod_mul

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'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` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.mod_mul`",
      "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"
    }
  ]
}