Mathlib v4.30 source proposition Nat.add_div_of_dvd_add_add_one

Identifier: F:ml430-nat-add-div-of-dvd-add-add-one-f17dffc0
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {c a b : ℕ}, c ∣ a + b + 1 → (a + b) / c = a / c + b / c

Dependencies

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

Evidence

kernel-Nat.add_div_of_dvd_add_add_one

Kind: kernel-term
Status: checked

Supports: ∀ {c a b : ℕ}, c ∣ a + b + 1 → (a + b) / c = a / c + b / c

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.add_div_of_dvd_add_add_one` through the trusted `Kernel::add_declaration` gate (declared in `nat_prelude/div_mod_lemmas.rs`'s `declare_add_div_of_dvd_add_add_one`, the ninth `ml430` add/div/mod mirror -- the eight in `declare_add_div_mod_shift_family` left this one open, per `docs/plan/status/283-nat-div-mod-family.md`). Route (see the module doc for the full derivation): decompose `a=c*qa+ra`, `b=c*qb+rb` via `div_mod_exec`; case-split `ra+rb+1` against `c` (`lt_or_ge`) -- below `c` this is already a valid `divMod` decomposition of `a+b+1` and comparing it against the `dvd`-witness relation (remainder `0`) via `div_mod_unique` forces `ra+rb+1=0`, refuted by `succ_ne_zero`; at or above `c`, subtracting `c` once (`sub_add_cancel`) gives a remainder `r'` bounded `<c` (via `ra<c`, `rb<c`, `le_of_succ_le_succ`/`add_le_add_left`/`add_le_add_right`/`le_trans`), and comparing THAT decomposition against the same `dvd`-witness relation forces `r'=0`, i.e. `ra+rb+1=c` exactly, pinning `ra+rb=c-1<c` -- which makes `(qa+qb, ra+rb)` a valid `divMod` decomposition of `a+b` itself, closed against `div_mod_exec`'s own decomposition of `a+b` via one more `div_mod_unique`. `nat_theorem_inventory`'s rendered type for `Nat.add_div_of_dvd_add_add_one` is `(x0:AxNat)->(x1:AxNat)->(x2:AxNat)->(x3:AxNat.dvd x0 (AxNat.add (AxNat.add x1 x2) (AxNat.succ AxNat.zero)))->Eq (AxNat.div (AxNat.add x1 x2) x0) (AxNat.add (AxNat.div x1 x0) (AxNat.div x2 x0))`, matching this fact's `formal.statement` verbatim (`x0`=c, `x1`=a, `x2`=b). `nat_theorem_inventory` exits non-zero for a name that does not exist (verified against `add_div_of_dvd_add_add_one_bogus`, count 0), and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed.

footprint-Nat.add_div_of_dvd_add_add_one

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 every individual theorem's 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), and the new `add_div_of_dvd_add_add_one_applies_at_concrete_discriminating_instances` test re-checks it plus two concrete instantiations chosen to discriminate: `(c,a,b)=(5,7,7)` (equal `a`,`b`, both quotients and both remainders nonzero, remainders summing exactly to `c-1`) and `(c,a,b)=(5,3,11)` (`a<c<=b`, asymmetric, to catch an `a`/`b` swap), both via `def_eq` against the independently-computed expected values.

Provenance

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