Mathlib v4.30 source proposition Int.add_le_add_iff_left

Identifier: F:ml430-int-add-le-add-iff-left-83bf44fa
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {b c : ℤ} (a : ℤ), a + b ≤ a + c ↔ b ≤ c

Dependencies

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

6 direct dependencies 0 direct dependents

Evidence

kernel-Int.add_le_add_iff_left

Kind: kernel-term
Status: checked

Supports: `Int.add_le_add_iff_left` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : Int) -> ((x1 : Int) -> ((x2 : Int) -> Iff (Int.le (Int.add x2 x0) (Int.add x2 x1)) (Int.le x0 x1))))`. Built in a new file, int_prelude/order_add.rs, as a corollary of the already-derived Int.add_le_add plus Int.add_neg_cancel_right (algebra.rs) and modeq.rs's cancel_neg_add/cancel_neg_add_left -- no Int.rec case split anywhere in the new file.

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.add_le_add_iff_left" && $4 == "((x0 : Int) -> ((x1 : Int) -> ((x2 : Int) -> Iff (Int.le (Int.add x2 x0) (Int.add x2 x1)) (Int.le x0 x1))))"' | wc -l)" -ge 1

footprint-Int.add_le_add_iff_left

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Int.add_le_add_iff_left`. 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.add_le_add_iff_left

Kind: kernel-term
Status: checked

Supports: Coverage for `Int.add_le_add_iff_left` 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-add (2026-08-29): built directly as a kernel declaration in the new file int_prelude/order_add.rs, as pure algebra on top of the already-derived Int.add_le_add (order.rs), Int.add_neg_cancel_right (algebra.rs), and modeq.rs's cancel_neg_add / cancel_neg_add_left (the latter widened from private to pub(super) for this reuse -- no other change to modeq.rs). No Int.rec case split anywhere in the new file.",
  "source": "statement-only extraction of `Int.add_le_add_iff_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.add_le_add_iff_left`",
      "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"
    }
  ]
}