Mathlib v4.30 source proposition Nat.mod_mul_right_mod

Identifier: F:ml430-nat-mod-mul-right-mod-a481eff8
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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

Evidence

kernel-Nat.mod_mul_right_mod

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

Declared in `nat_prelude/mod_mul_lemmas.rs`'s `declare_mod_mul_family`. Case-splits `b` (outer `cases_zero_succ`): at `b=0` both sides collapse to `mod a zero` via `zero_mul`/`mod_zero` congruence (a single `mod_zero` applied to the already-doubled `mod (mod a zero) zero` term, since `dvsr` and the outer divisor coincide at `zero` here); at `b=succ bpred`, case-splits `c`: at `c=0` both sides collapse to `a%b` via `mul_zero`/`mod_zero`; at `c=succ cpred` (both positive), the same `mod_of_dvd_mod` helper as `mod_mul_left_mod` answers the goal directly with `dvsr:=b`, `mult:=c`, `e:=b*c` -- no `mul_comm` bridge needed here since `e` is already `dvsr*mult` in the fact's own order (`e_eq` is `refl`). `nat_theorem_inventory`'s rendered type matches this fact's `formal.statement` verbatim (`x0`=a, `x1`=b, `x2`=c).

footprint-Nat.mod_mul_right_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_right_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_right_mod` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.mod_mul_right_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"
    }
  ]
}