PD0006

IsGCD(g,a,b)

g is a common divisor divisible by every common divisor.

Conservative notation; not a theorem, primitive, or axiom.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Hygienic expanded first-order definition

((exists x. a = g * x) /\ (exists y. b = g * y)) /\ forall d. (exists u. a = d * u) -> (exists v. b = d * v) -> exists w. g = d * w

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Dvd(d,n)

Definitions depending on this notation

CRTMergeCompatiblePrefix(r,s,b,c,l)CRTPairwiseCompatiblePrefix(r,s,b,c,l)

Checked theorems using this definition

GC0006 · crt_pairwise_compatible_prefix_lastGC0008 · generalized_binary_crt_merge_stepGC000E · crt_is_gcd_scaleGC000F · crt_is_gcd_coprime_factor_removeGC0011 · crt_is_gcd_coprime_productGC0012 · crt_lcm_gcd_cofactor_productGC0013 · crt_gcd_scaled_coprime_componentGC0014 · crt_gcd_monotone_under_divisibilityGC0015 · crt_gcd_lcm_distributes_divisibility

Separate complete second-wave branches: Full G011 proof · Alpha v27.