Mathlib v4.30 source proposition Int.le_of_ofNat_le_ofNat

Identifier: F:ml430-int-le-of-ofnat-le-ofnat-483e0f18
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.le_of_ofNat_le_ofNat` in the pinned Mathlib v4.30 source.

Formal statement

∀ {m n : ℕ}, ↑m ≤ ↑n → m ≤ n

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Int.le_of_ofNat_le_ofNat

Kind: kernel-term
Status: checked

Supports: `Int.le_of_ofNat_le_ofNat` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Int.le (Int.ofNat x0) (Int.ofNat x1)) -> AxNat.le x0 x1)))`. `Int.le`/`Int.lt` are built by `define_binary_int` (int_prelude/defs.rs), whose `ofNat`/`ofNat` branch is literally `NatOps::le`/`NatOps::lt` on the two `Nat` fields, so `Int.le (ofNat m) (ofNat n)` is definitionally `Nat.le m n` and the proof is the hypothesis itself under that defeq -- no case split, no lemma. Built in a new file, int_prelude/order_coercion.rs.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example int_theorem_inventory 2>/dev/null | awk -F'\t' '$1 == "theorem" && $2 == "Int.le_of_ofNat_le_ofNat" && $4 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Int.le (Int.ofNat x0) (Int.ofNat x1)) -> AxNat.le x0 x1)))"' | wc -l)" -ge 1

footprint-Int.le_of_ofNat_le_ofNat

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Int.le_of_ofNat_le_ofNat`. The `integer` prelude's trusted surface stays 0: no Axiom, no Opaque, no Quotient.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free integer

coverage-Int.le_of_ofNat_le_ofNat

Kind: kernel-term
Status: checked

Supports: Coverage for `Int.le_of_ofNat_le_ofNat` derived from `kernel.environment()` directly, not from an inventory list -- an Int-namespace declaration missing from `derived_laws` fails this test by name.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::every_int_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 the new file int_prelude/order_coercion.rs. Purely definitional -- no Int.rec case split, no dependency on any other derived law.",
  "source": "statement-only extraction of `Int.le_of_ofNat_le_ofNat` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.le_of_ofNat_le_ofNat`",
      "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"
    }
  ]
}