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_u_ph_explorer. ∃ ff_v_ph_explorer. Beta(ff_u_ph_explorer,ff_v_ph_explorer,0,0) ∧ (Beta(ff_u_ph_explorer,ff_v_ph_explorer,ell,z) ∧ (∀ y. Lt(y,ell) → ∃ n. ∃ m. ∃ k. Beta(b,c,y,n) ∧ (Beta(ff_u_ph_explorer,ff_v_ph_explorer,y,m) ∧ (Beta(ff_u_ph_explorer,ff_v_ph_explorer,S y,k) ∧ k = m · x + n))))Only definitions earlier in this acyclic notation graph are used here.
Hygienic expanded first-order definition
exists ff_u_ph_explorer ff_v_ph_explorer. ((((exists fs_h_ph_explorer_body_start. fs_h_ph_explorer_body_start + S (0) = S ((S (0)) * ff_v_ph_explorer)) /\ exists fs_q_ph_explorer_body_start. ff_u_ph_explorer = fs_q_ph_explorer_body_start * S ((S (0)) * ff_v_ph_explorer) + (0))) /\ ((((exists fs_h_ph_explorer_body_terminal. fs_h_ph_explorer_body_terminal + S (z) = S ((S (ell)) * ff_v_ph_explorer)) /\ exists fs_q_ph_explorer_body_terminal. ff_u_ph_explorer = fs_q_ph_explorer_body_terminal * S ((S (ell)) * ff_v_ph_explorer) + (z))) /\ forall ff_i_ph_explorer_body_steps. (exists ph_bound_explorer_body_steps. ph_bound_explorer_body_steps + S ff_i_ph_explorer_body_steps = ell) -> exists ff_coefficient_ph_explorer_body_steps ff_previous_ph_explorer_body_steps ff_current_ph_explorer_body_steps. ((((exists fs_h_ph_explorer_body_steps_coefficient. fs_h_ph_explorer_body_steps_coefficient + S (ff_coefficient_ph_explorer_body_steps) = S ((S (ff_i_ph_explorer_body_steps)) * c)) /\ exists fs_q_ph_explorer_body_steps_coefficient. b = fs_q_ph_explorer_body_steps_coefficient * S ((S (ff_i_ph_explorer_body_steps)) * c) + (ff_coefficient_ph_explorer_body_steps))) /\ ((((exists fs_h_ph_explorer_body_steps_before. fs_h_ph_explorer_body_steps_before + S (ff_previous_ph_explorer_body_steps) = S ((S (ff_i_ph_explorer_body_steps)) * ff_v_ph_explorer)) /\ exists fs_q_ph_explorer_body_steps_before. ff_u_ph_explorer = fs_q_ph_explorer_body_steps_before * S ((S (ff_i_ph_explorer_body_steps)) * ff_v_ph_explorer) + (ff_previous_ph_explorer_body_steps))) /\ ((((exists fs_h_ph_explorer_body_steps_after. fs_h_ph_explorer_body_steps_after + S (ff_current_ph_explorer_body_steps) = S ((S (S ff_i_ph_explorer_body_steps)) * ff_v_ph_explorer)) /\ exists fs_q_ph_explorer_body_steps_after. ff_u_ph_explorer = fs_q_ph_explorer_body_steps_after * S ((S (S ff_i_ph_explorer_body_steps)) * ff_v_ph_explorer) + (ff_current_ph_explorer_body_steps))) /\ ff_current_ph_explorer_body_steps = ff_previous_ph_explorer_body_steps * x + ff_coefficient_ph_explorer_body_steps)))))The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.