Mathlib v4.30 source proposition Int.add_neg_eq_sub

Identifier: F:ml430-int-add-neg-eq-sub-32f34729
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b : ℤ}, a + -b = a - b

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-Int.add_neg_eq_sub

Kind: kernel-term
Status: checked

Supports: `Int.add_neg_eq_sub` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : Int) -> ((x1 : Int) -> Eq.{1} Int (Int.add x0 (Int.neg x1)) (Int.sub x0 x1)))`.

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_neg_eq_sub" && $4 == "((x0 : Int) -> ((x1 : Int) -> Eq.{1} Int (Int.add x0 (Int.neg x1)) (Int.sub x0 x1)))"' | wc -l)" -ge 1

footprint-Int.add_neg_eq_sub

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Int.add_neg_eq_sub`. 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_neg_eq_sub

Kind: kernel-term
Status: checked

Supports: Coverage for `Int.add_neg_eq_sub` 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-add-basics (2026-08-29): built directly as a kernel declaration in the new file int_prelude/add_basics.rs, as pure algebra on top of the already-derived Int.add_comm, Int.add_assoc, Int.add_zero, Int.add_neg, Int.mul_comm, Int.left_distrib (algebra.rs), Int.sub's definition (sub.rs), and modeq.rs's private cancel_neg_add_left (already pub(super) for order_add.rs's reuse). No Int.rec case split anywhere in the new file.",
  "source": "statement-only extraction of `Int.add_neg_eq_sub` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.add_neg_eq_sub`",
      "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"
    }
  ]
}