Mathlib v4.30 source proposition Int.le.elim

Identifier: F:ml430-int-le-elim-efa70bfa
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b : ℤ}, a ≤ b → ∀ {P : Prop}, (∀ (n : ℕ), a + ↑n = b → P) → P

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Int.le.elim

Kind: kernel-term
Status: checked

Supports: `Int.le.elim` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : Int) -> ((x1 : Int) -> ((x2 : Int.le x0 x1) -> ((x3 : Prop) -> ((x4 : ((x4 : AxNat) -> ((x5 : Eq.{1} Int (Int.add x0 (Int.ofNat x4)) x1) -> x3))) -> x3)))))`. This is the CPS elimination form of `Int.le_dest`'s `Exists` witness -- `le_dest` produces `b = a + ofNat i`, Mathlib's direction is `a + ofNat n = b`, so the equation is flipped with `isymm` before handing it to the minor premise. Declared as a child of `le` (`Int.le.elim`), the same namespacing `Nat.le.step` uses for a name under an unrelated head. Only the Exists-flavoured cousin (`Int.le_dest`) existed under any name before this -- confirmed by a shape-search returning nothing at >=0.75 for the CPS shape. Built via `Exists.elim` (ops::exists_elim) 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.elim" && $4 == "((x0 : Int) -> ((x1 : Int) -> ((x2 : Int.le x0 x1) -> ((x3 : Prop) -> ((x4 : ((x4 : AxNat) -> ((x5 : Eq.{1} Int (Int.add x0 (Int.ofNat x4)) x1) -> x3))) -> x3)))))"' | wc -l)" -ge 1

footprint-Int.le.elim

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Int.le.elim`. 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.elim

Kind: kernel-term
Status: checked

Supports: Coverage for `Int.le.elim` 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, as Exists.elim applied to the already-proved F:int-le-dest with the witness equation flipped by isymm.",
  "source": "statement-only extraction of `Int.le.elim` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.le.elim`",
      "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"
    }
  ]
}