Mathlib v4.30 source proposition Nat.add_eq_two_iff

Identifier: F:ml430-nat-add-eq-two-iff-25385c65
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m + n = 2 ↔ m = 0 ∧ n = 2 ∨ m = 1 ∧ n = 1 ∨ m = 2 ∧ n = 0

Dependencies

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

6 direct dependencies 0 direct dependents

Evidence

kernel-Nat.add_eq_two_iff

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, m + n = 2 ↔ m = 0 ∧ n = 2 ∨ m = 1 ∧ n = 1 ∨ m = 2 ∧ n = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.add_eq_two_iff` through the trusted `Kernel::add_declaration` gate, declared in `nat_prelude/add_basics.rs`. Declared by `declare_add_eq_lit_iff(d, p, p.add_eq_two_iff, 2)`, the shared helper for the whole `add_eq_{one,two,three}_iff` group (`nat_prelude/add_basics.rs`). `mp` bounds `m <= 2` (`le_add_right` transported along the hypothesis), then walks `lt_or_eq_of_le` / `le_of_lt_succ` from bound 2 down to 0, closing each `Eq` leaf by recovering `n` via `add_left_cancel` and placing the resulting `And` into the right-associated `Or` at the matching position; the final `Lt m 0` leaf is a contradiction via `not_lt_zero`. `mpr` walks the same `Or` shape via a private `or_elim` and closes each branch's concrete arithmetic identity by `Eq.refl` (small numerals fully reduce by defeq). `nat_theorem_inventory`'s rendered type is `(x0,x1:AxNat)->Iff (Eq (add x0 x1) 2) (Or (And 0 2) (Or (And 1 1) (And 2 0)))`, matching this fact's `formal.statement`. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Verified both ways: the real name greps to a count `-ge 1`; grepping a made-up name (`Nat.add_eq_two_iff_bogus`) greps to `0`.

footprint-Nat.add_eq_two_iff

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`, and `the_build_is_deterministic`/`every_promised_name_is_admitted_with_the_expected_kind` (both re-run on every `nat_prelude::` sweep) confirm the declaration is admitted at the exact rendered type checked below.

Provenance

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