Mathlib v4.30 source proposition Nat.mod_mul_left_mod

Identifier: F:ml430-nat-mod-mul-left-mod-9b785abc
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (a b c : ℕ), a % (b * c) % c = a % c

Dependencies

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

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

Evidence

kernel-Nat.mod_mul_left_mod

Kind: kernel-term
Status: checked

Supports: ∀ (a b c : ℕ), a % (b * c) % c = a % c

Checker command

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

Evidence notes

Declared in `nat_prelude/mod_mul_lemmas.rs`'s `declare_mod_mul_family`. Case-splits `c` (outer `cases_zero_succ`): at `c=0` both sides collapse to `a` via `mul_zero`/`mod_zero` congruence; at `c=succ cpred`, case-splits `b`: at `b=0` both sides collapse to `a%c` via `zero_mul`/`mod_zero`; at `b=succ bpred` (both positive), the general local helper `mod_of_dvd_mod` answers `a%(dvsr*mult)%dvsr = a%dvsr` for any multiple `dvsr*mult` of the divisor `dvsr` -- decompose `a` at `e:=b*c` to get `a=e*qe+re`, decompose the remainder `re` at `c` to get `re=c*qd+rd`, substitute to get a second valid `divMod c a _ rd` decomposition, and `div_mod_unique` against the canonical one forces `rd = a%c`; `rd` is literally `mod (mod a e) c` by construction, which is the goal (`e := b*c`, bridged to `c*b` via `mul_comm` since `mod_of_dvd_mod`'s hypothesis wants `e = dvsr*mult`). `nat_theorem_inventory`'s rendered type matches this fact's `formal.statement` verbatim (`x0`=a, `x1`=b, `x2`=c).

footprint-Nat.mod_mul_left_mod

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