Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Definition in prerequisite notation
¬n = 0 ∧ (Product(b,c,l,n) ∧ AllPrime(b,c,l))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
(~(n = 0) /\ ((exists ff_u_fsat_lowerlayer_product ff_v_fsat_lowerlayer_product. ((((exists ff_h_fsat_lowerlayer_product_start. ff_h_fsat_lowerlayer_product_start + S (1) = S ((S (0)) * ff_v_fsat_lowerlayer_product)) /\ exists ff_q_fsat_lowerlayer_product_start. ff_u_fsat_lowerlayer_product = ff_q_fsat_lowerlayer_product_start * S ((S (0)) * ff_v_fsat_lowerlayer_product) + (1))) /\ ((((exists ff_h_fsat_lowerlayer_product_terminal. ff_h_fsat_lowerlayer_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_lowerlayer_product)) /\ exists ff_q_fsat_lowerlayer_product_terminal. ff_u_fsat_lowerlayer_product = ff_q_fsat_lowerlayer_product_terminal * S ((S (l)) * ff_v_fsat_lowerlayer_product) + (n))) /\ forall ff_i_fsat_lowerlayer_product. (exists ff_lt_fsat_lowerlayer_product_bound. ff_lt_fsat_lowerlayer_product_bound + S ff_i_fsat_lowerlayer_product = l) -> exists ff_p_fsat_lowerlayer_product ff_r_fsat_lowerlayer_product ff_s_fsat_lowerlayer_product. ((((exists ff_h_fsat_lowerlayer_product_factor. ff_h_fsat_lowerlayer_product_factor + S (ff_p_fsat_lowerlayer_product) = S ((S (ff_i_fsat_lowerlayer_product)) * c)) /\ exists ff_q_fsat_lowerlayer_product_factor. b = ff_q_fsat_lowerlayer_product_factor * S ((S (ff_i_fsat_lowerlayer_product)) * c) + (ff_p_fsat_lowerlayer_product))) /\ ((((exists ff_h_fsat_lowerlayer_product_partial. ff_h_fsat_lowerlayer_product_partial + S (ff_r_fsat_lowerlayer_product) = S ((S (ff_i_fsat_lowerlayer_product)) * ff_v_fsat_lowerlayer_product)) /\ exists ff_q_fsat_lowerlayer_product_partial. ff_u_fsat_lowerlayer_product = ff_q_fsat_lowerlayer_product_partial * S ((S (ff_i_fsat_lowerlayer_product)) * ff_v_fsat_lowerlayer_product) + (ff_r_fsat_lowerlayer_product))) /\ ((((exists ff_h_fsat_lowerlayer_product_successor. ff_h_fsat_lowerlayer_product_successor + S (ff_s_fsat_lowerlayer_product) = S ((S (S ff_i_fsat_lowerlayer_product)) * ff_v_fsat_lowerlayer_product)) /\ exists ff_q_fsat_lowerlayer_product_successor. ff_u_fsat_lowerlayer_product = ff_q_fsat_lowerlayer_product_successor * S ((S (S ff_i_fsat_lowerlayer_product)) * ff_v_fsat_lowerlayer_product) + (ff_s_fsat_lowerlayer_product))) /\ ff_s_fsat_lowerlayer_product = ff_r_fsat_lowerlayer_product * ff_p_fsat_lowerlayer_product)))))) /\ (forall ftsf_index_fsat_lowerlayer_primes. (exists ftsf_gap_fsat_lowerlayer_primes_bound. ftsf_gap_fsat_lowerlayer_primes_bound + S ftsf_index_fsat_lowerlayer_primes = (l)) -> exists ftsf_factor_fsat_lowerlayer_primes. ((((exists ff_h_ftsf_fsat_lowerlayer_primes_entry. ff_h_ftsf_fsat_lowerlayer_primes_entry + S (ftsf_factor_fsat_lowerlayer_primes) = S ((S (ftsf_index_fsat_lowerlayer_primes)) * c)) /\ exists ff_q_ftsf_fsat_lowerlayer_primes_entry. b = ff_q_ftsf_fsat_lowerlayer_primes_entry * S ((S (ftsf_index_fsat_lowerlayer_primes)) * c) + (ftsf_factor_fsat_lowerlayer_primes))) /\ ((~(ftsf_factor_fsat_lowerlayer_primes = 1) /\ forall frm_prime_left_ftsf_fsat_lowerlayer_primes_prime frm_prime_right_ftsf_fsat_lowerlayer_primes_prime. ftsf_factor_fsat_lowerlayer_primes = frm_prime_left_ftsf_fsat_lowerlayer_primes_prime * frm_prime_right_ftsf_fsat_lowerlayer_primes_prime -> frm_prime_left_ftsf_fsat_lowerlayer_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_lowerlayer_primes_prime = 1))))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.