Mathlib v4.30 source proposition Nat.size_le_size

Identifier: F:ml430-nat-size-le-size-c4b98f53
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m n : ℕ}, m ≤ n → m.size ≤ n.size

Dependencies

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

0 direct dependencies 0 direct dependents

Evidence

kernel-Nat.size_le_size

Kind: kernel-term
Status: checked

Supports: ∀ {m n : ℕ}, m ≤ n → m.size ≤ n.size

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.size_le_size` through the trusted `Kernel::add_declaration` gate, declared in the new module `nat_prelude/size_order.rs`. `nat_theorem_inventory`'s rendered type is `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le x0 x1) -> AxNat.le (AxNat.size x0) (AxNat.size x1))))`, 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.size_le_size_bogus`) greps to `0`.

footprint-Nat.size_le_size

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. A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds this 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` (this declaration's `p.size_le_size` entry was added there in the same session that landed it), and `size_order_tests::size_le_size_applies_at_a_concrete_pair_and_symbolically` independently checks `axiom_footprint(p.size_le_size).is_empty()`, exercises the theorem at a concrete instance (`m := 3, n := 6`: `size 3 = 2`, `size 6 = 3`, `2 <= 3`) and at a fully symbolic (no numerals at all) instance, plus a negative control (the statement must not also equal the reversed inequality `Le (size 6) (size 3)`).

Provenance

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