Mathlib v4.30 source proposition Nat.add_pos_right

Identifier: F:ml430-nat-add-pos-right-e43374dc
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {b : ℕ} (a : ℕ), 0 < b → 0 < a + b

Dependencies

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

3 direct dependencies 2 direct dependents

Evidence

kernel-Nat.add_pos_right

Kind: kernel-term
Status: checked

Supports: `Nat.add_pos_right` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `Nat.add_pos_right\t3\t((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.lt AxNat.zero x0) -> AxNat.lt AxNat.zero (AxNat.add x1 x0))))` (binders `b a` -- Mathlib's `{b}`-implicit then explicit `a`; content-identical, argument order not observable in a Prop). A case split on `b` (NatOps::induct, ih unused): at `zero` the hypothesis `Lt zero zero` is impossible, discharged by `Nat.not_lt_zero` at `zero`; at `succ k`, `add a (succ k)` is definitionally `succ (add a k)` (`Nat.add` recurses on its RIGHT argument), so the conclusion is exactly `NatOps::zero_lt_succ` applied to `add a k`, independent of the hypothesis. Built in a new file, nat_prelude/add_pos.rs.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory -- add_pos_right 2>/dev/null | grep -xFc $'Nat.add_pos_right\t3\t((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.lt AxNat.zero x0) -> AxNat.lt AxNat.zero (AxNat.add x1 x0))))')" -ge 1

footprint-Nat.add_pos_right

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] for `Nat.add_pos_right`. The `nat` prelude's trusted surface stays 0: no Axiom, no Opaque, no Quotient.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example theorem_axiom_footprint -- Nat.add_pos_right 2>/dev/null | grep -xFc $'nat\tNat.add_pos_right\t0\t')" -ge 1

coverage-Nat.add_pos_right

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.add_pos_right` derived from `kernel.environment()` directly, not from an inventory list -- a Nat-namespace declaration missing from `theorem_names` fails this test by name.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Provenance

{
  "date": "2026-08-29",
  "established_by": "lane int-order-coercion (2026-08-30): built directly as a kernel declaration in a new file, nat_prelude/add_pos.rs. Same shape as nat_prelude/order_more.rs's zero_lt_of_ne_zero (a case split via NatOps::induct with the ih unused); no dependency on any other derived law.",
  "source": "statement-only extraction of `Nat.add_pos_right` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.add_pos_right`",
      "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"
    }
  ]
}