Mathlib v4.30 source proposition Int.dvd_gcd_mul_gcd_iff_dvd_mul

build_int_prelude admits Int.dvd_gcd_mul_gcd_iff_dvd_mul through the trusted Kernel::add_declaration gate, so producing this row at all is a machine-checked proof. New proof, lane int-gcd-mul-transport: int_prelude/gcd_scaled_mirrors.rs. Built by applying the shared shape at (b,c) := (n, ofNat (k.gcd m)), then chaining one more Iff.trans against the already-proved Int.dvd_mul_gcd_iff_dvd_mul, mirroring nat_prelude/gcd_mul_right_mirrors.rs's dvd_gcd_mul_gcd_iff_dvd_mul one layer up (which is why dvd_mul_gcd_iff_dvd_mul had to be declared first). int_theorem_inventory's rendered type is `(x0:Int)->(x1:Int)->(x2:Int)->Iff (Int.dvd x0 (Int.mul (Int.ofNat (Int.gcd x0 x1)) (Int.ofNat (Int.gcd x0 x2)))) (Int.dvd x0 (Int.mul x1 x2))`, matching this fact's formal.statement character for character (x0/x1/x2 = k/n/m). int_theorem_inventory exits non-zero for a name that does not exist (verified: dvd_gcd_mul_gcd_iff_dvd_mul_bogus_xyz -> exit 1, "no Int declaration matches"), and the anchored grep -c (tested -ge 1, not piped through grep -q) requires the exact name followed by whitespace, so it does not also match the substring-overlapped Int.dvd_gcd_mul_iff_dvd_mul.

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.