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

EuclideanAnchoredExecution(a,b,g,k)EuclideanExecution(a,b,g,k)

Checked theorems using this definition

GT0006 · euclidean_gcd_step_forwardGT0007 · euclidean_gcd_step_backwardGT0008 · euclidean_gcd_step_iffGT0009 · euclidean_gcd_step_output_uniqueGT000A · euclidean_gcd_zero_terminal_uniqueGT000D · euclidean_trace_prefix_gcd_invariantGT000E · euclidean_trace_initial_state_is_gcdGT000F · euclidean_trace_terminal_gcd_existsGT0013 · euclidean_anchored_execution_gcd_correct