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.
Definition in prerequisite notation
(∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → Dvd(g,y)) ∧ (∀ x. (∀ y. ∀ z. Lt(y,l) → BetaAt(b,c,y,z) → Dvd(x,z)) → Dvd(x,g))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
((forall ppf_index_prioritylayercommon ppf_entry_prioritylayercommon. (exists pvs_gap_prioritylayercommonbound. pvs_gap_prioritylayercommonbound + S (ppf_index_prioritylayercommon) = ((l))) -> (((exists ff_h_pvs_prioritylayercommonentry. ff_h_pvs_prioritylayercommonentry + S (ppf_entry_prioritylayercommon) = S ((S (ppf_index_prioritylayercommon)) * (c))) /\ exists ff_q_pvs_prioritylayercommonentry. (b) = ff_q_pvs_prioritylayercommonentry * S ((S (ppf_index_prioritylayercommon)) * (c)) + (ppf_entry_prioritylayercommon))) -> (exists pvs_factor_prioritylayercommondivisor. (ppf_entry_prioritylayercommon) = ((g)) * pvs_factor_prioritylayercommondivisor)) /\ (forall ppf_common_prioritylayer. (forall ppf_index_prioritylayerother ppf_entry_prioritylayerother. (exists pvs_gap_prioritylayerotherbound. pvs_gap_prioritylayerotherbound + S (ppf_index_prioritylayerother) = ((l))) -> (((exists ff_h_pvs_prioritylayerotherentry. ff_h_pvs_prioritylayerotherentry + S (ppf_entry_prioritylayerother) = S ((S (ppf_index_prioritylayerother)) * (c))) /\ exists ff_q_pvs_prioritylayerotherentry. (b) = ff_q_pvs_prioritylayerotherentry * S ((S (ppf_index_prioritylayerother)) * (c)) + (ppf_entry_prioritylayerother))) -> (exists pvs_factor_prioritylayerotherdivisor. (ppf_entry_prioritylayerother) = (ppf_common_prioritylayer) * pvs_factor_prioritylayerotherdivisor)) -> (exists pvs_factor_prioritylayergreatest. ((g)) = (ppf_common_prioritylayer) * pvs_factor_prioritylayergreatest)))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.
Direct definition dependencies
Definitions depending on this notation
Checked theorems using this definition
SK001E · prime_exponent_prefix_gcd_emptySK001F · prime_exponent_prefix_gcd_successorSK0020 · prime_exponent_prefix_gcd_existsSK0021 · prime_exponent_prefix_gcd_functionalSK0023 · prime_valuation_support_exponent_gcd_nonzeroSK0027 · prime_support_exponent_gcd_divisor_criterionSK0028 · prime_support_perfect_power_iff_degree_dividesSK0029 · prime_support_exponent_gcd_roots_availableSK002F · perfect_power_profile_exists