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
∃ ff_code_dot_explorer. ∃ ff_scale_dot_explorer. (∀ x. ∀ y. ∀ n. ∀ m. Lt(x,ell) → Beta(b,c,x,y) → Beta(d,e,x,n) → Beta(ff_code_dot_explorer,ff_scale_dot_explorer,x,m) → m = y · n) ∧ Sum(ff_code_dot_explorer,ff_scale_dot_explorer,ell,z)Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists ff_code_dot_explorer ff_scale_dot_explorer. ((forall fpmp_index_dot_explorer_pointwise fpmp_left_dot_explorer_pointwise fpmp_right_dot_explorer_pointwise fpmp_target_dot_explorer_pointwise. (exists fpmp_gap_dot_explorer_pointwise. fpmp_gap_dot_explorer_pointwise + S fpmp_index_dot_explorer_pointwise = ell) -> (((exists ff_h_fpmp_dot_explorer_pointwise_left. ff_h_fpmp_dot_explorer_pointwise_left + S (fpmp_left_dot_explorer_pointwise) = S ((S (fpmp_index_dot_explorer_pointwise)) * c)) /\ exists ff_q_fpmp_dot_explorer_pointwise_left. b = ff_q_fpmp_dot_explorer_pointwise_left * S ((S (fpmp_index_dot_explorer_pointwise)) * c) + (fpmp_left_dot_explorer_pointwise))) -> (((exists ff_h_fpmp_dot_explorer_pointwise_right. ff_h_fpmp_dot_explorer_pointwise_right + S (fpmp_right_dot_explorer_pointwise) = S ((S (fpmp_index_dot_explorer_pointwise)) * e)) /\ exists ff_q_fpmp_dot_explorer_pointwise_right. d = ff_q_fpmp_dot_explorer_pointwise_right * S ((S (fpmp_index_dot_explorer_pointwise)) * e) + (fpmp_right_dot_explorer_pointwise))) -> (((exists ff_h_fpmp_dot_explorer_pointwise_target. ff_h_fpmp_dot_explorer_pointwise_target + S (fpmp_target_dot_explorer_pointwise) = S ((S (fpmp_index_dot_explorer_pointwise)) * ff_scale_dot_explorer)) /\ exists ff_q_fpmp_dot_explorer_pointwise_target. ff_code_dot_explorer = ff_q_fpmp_dot_explorer_pointwise_target * S ((S (fpmp_index_dot_explorer_pointwise)) * ff_scale_dot_explorer) + (fpmp_target_dot_explorer_pointwise))) -> fpmp_target_dot_explorer_pointwise = fpmp_left_dot_explorer_pointwise * fpmp_right_dot_explorer_pointwise) /\ (exists ff_u_dot_explorer_sum ff_v_dot_explorer_sum. ((((exists ff_h_dot_explorer_sum_start. ff_h_dot_explorer_sum_start + S (0) = S ((S (0)) * ff_v_dot_explorer_sum)) /\ exists ff_q_dot_explorer_sum_start. ff_u_dot_explorer_sum = ff_q_dot_explorer_sum_start * S ((S (0)) * ff_v_dot_explorer_sum) + (0))) /\ ((((exists ff_h_dot_explorer_sum_terminal. ff_h_dot_explorer_sum_terminal + S (z) = S ((S (ell)) * ff_v_dot_explorer_sum)) /\ exists ff_q_dot_explorer_sum_terminal. ff_u_dot_explorer_sum = ff_q_dot_explorer_sum_terminal * S ((S (ell)) * ff_v_dot_explorer_sum) + (z))) /\ forall ff_i_dot_explorer_sum. (exists ff_lt_dot_explorer_sum_bound. ff_lt_dot_explorer_sum_bound + S ff_i_dot_explorer_sum = ell) -> exists ff_a_dot_explorer_sum ff_r_dot_explorer_sum ff_s_dot_explorer_sum. ((((exists ff_h_dot_explorer_sum_summand. ff_h_dot_explorer_sum_summand + S (ff_a_dot_explorer_sum) = S ((S (ff_i_dot_explorer_sum)) * ff_scale_dot_explorer)) /\ exists ff_q_dot_explorer_sum_summand. ff_code_dot_explorer = ff_q_dot_explorer_sum_summand * S ((S (ff_i_dot_explorer_sum)) * ff_scale_dot_explorer) + (ff_a_dot_explorer_sum))) /\ ((((exists ff_h_dot_explorer_sum_partial. ff_h_dot_explorer_sum_partial + S (ff_r_dot_explorer_sum) = S ((S (ff_i_dot_explorer_sum)) * ff_v_dot_explorer_sum)) /\ exists ff_q_dot_explorer_sum_partial. ff_u_dot_explorer_sum = ff_q_dot_explorer_sum_partial * S ((S (ff_i_dot_explorer_sum)) * ff_v_dot_explorer_sum) + (ff_r_dot_explorer_sum))) /\ ((((exists ff_h_dot_explorer_sum_successor. ff_h_dot_explorer_sum_successor + S (ff_s_dot_explorer_sum) = S ((S (S ff_i_dot_explorer_sum)) * ff_v_dot_explorer_sum)) /\ exists ff_q_dot_explorer_sum_successor. ff_u_dot_explorer_sum = ff_q_dot_explorer_sum_successor * S ((S (S ff_i_dot_explorer_sum)) * ff_v_dot_explorer_sum) + (ff_s_dot_explorer_sum))) /\ ff_s_dot_explorer_sum = ff_r_dot_explorer_sum + ff_a_dot_explorer_sum)))))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.