Mathlib v4.30 source proposition Int.ModEq.eq

`build_int_prelude` admits `Int.mod_eq_emod_eq` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. New proof, lane int-dvd-mirrors: `int_prelude/dvd_gcd_mirrors.rs`'s `declare_mod_eq_emod_eq` -- Mathlib's `Int.ModEq.eq`, UNCONDITIONAL in `n`. `Int.ModEq n a b` IS DEFINED as `Eq Int (emod a n) (emod b n)` (`modeq.rs`'s `declare_modeq_definition`): the hypothesis already has exactly the goal's type up to unfolding that one `Definition`, so the proof is the hypothesis itself. `int_theorem_inventory`'s rendered type matches this fact's `formal.statement` (`∀ {n a b : ℤ}, a ≡ b [ZMOD n] → a % n = b % n`). `int_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; the anchor requires the exact name followed by whitespace, so it cannot match a longer sibling name sharing the same prefix. Verified both ways: the real name greps to a count `-ge 1`; grepping a fabricated name (`Int.mod_eq_emod_eq_bogus_xyz`) makes `int_theorem_inventory` fail closed (exit 1, "no Int declaration matches").

`prelude_axiom_inventory --require-axiom-free integer` enumerates the built Int environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: integer axiom=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by []. int_prelude_tests::every_int_declaration_is_checked_and_axiom_free and derived_laws_have_no_axiom_footprint (both re-run on every int_prelude:: sweep) additionally confirm this theorem's own Kernel::axiom_footprint directly.