square_factor_even_valuation_reflects_cofactor

Statement with defined notation

forall p z n e f g. ((~(p = 1) /\ forall frm_prime_left_ftsp_p frm_prime_right_ftsp_p. p = frm_prime_left_ftsp_p * frm_prime_right_ftsp_p -> frm_prime_left_ftsp_p = 1 \/ frm_prime_right_ftsp_p = 1)) -> ~(z = 0) -> ~(n = 0) -> (((exists bpv_gap_ftsp_square_factor_exponent_bound. bpv_gap_ftsp_square_factor_exponent_bound + e = z) /\ (exists bpv_result_ftsp_square_factor_selected. ((exists ff_b_ftsp_square_factor_selected_power ff_c_ftsp_square_factor_selected_power. ((forall ff_i_ftsp_square_factor_selected_power_repeat. (exists ff_lt_ftsp_square_factor_selected_power_repeat_bound. ff_lt_ftsp_square_factor_selected_power_repeat_bound + S ff_i_ftsp_square_factor_selected_power_repeat = e) -> (((exists ff_h_ftsp_square_factor_selected_power_repeat_decoded. ff_h_ftsp_square_factor_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsp_square_factor_selected_power_repeat)) * ff_c_ftsp_square_factor_selected_power)) /\ exists ff_q_ftsp_square_factor_selected_power_repeat_decoded. ff_b_ftsp_square_factor_selected_power = ff_q_ftsp_square_factor_selected_power_repeat_decoded * S ((S (ff_i_ftsp_square_factor_selected_power_repeat)) * ff_c_ftsp_square_factor_selected_power) + (p)))) /\ (exists ff_u_ftsp_square_factor_selected_power_product ff_v_ftsp_square_factor_selected_power_product. ((((exists ff_h_ftsp_square_factor_selected_power_product_start. ff_h_ftsp_square_factor_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsp_square_factor_selected_power_product)) /\ exists ff_q_ftsp_square_factor_selected_power_product_start. ff_u_ftsp_square_factor_selected_power_product = ff_q_ftsp_square_factor_selected_power_product_start * S ((S (0)) * ff_v_ftsp_square_factor_selected_power_product) + (1))) /\ ((((exists ff_h_ftsp_square_factor_selected_power_product_terminal. ff_h_ftsp_square_factor_selected_power_product_terminal + S (bpv_result_ftsp_square_factor_selected) = S ((S (e)) * ff_v_ftsp_square_factor_selected_power_product)) /\ exists ff_q_ftsp_square_factor_selected_power_product_terminal. ff_u_ftsp_square_factor_selected_power_product = ff_q_ftsp_square_factor_selected_power_product_terminal * S ((S (e)) * ff_v_ftsp_square_factor_selected_power_product) + (bpv_result_ftsp_square_factor_selected))) /\ forall ff_i_ftsp_square_factor_selected_power_product. (exists ff_lt_ftsp_square_factor_selected_power_product_bound. ff_lt_ftsp_square_factor_selected_power_product_bound + S ff_i_ftsp_square_factor_selected_power_product = e) -> exists ff_p_ftsp_square_factor_selected_power_product ff_r_ftsp_square_factor_selected_power_product ff_s_ftsp_square_factor_selected_power_product. ((((exists ff_h_ftsp_square_factor_selected_power_product_factor. ff_h_ftsp_square_factor_selected_power_product_factor + S (ff_p_ftsp_square_factor_selected_power_product) = S ((S (ff_i_ftsp_square_factor_selected_power_product)) * ff_c_ftsp_square_factor_selected_power)) /\ exists ff_q_ftsp_square_factor_selected_power_product_factor. ff_b_ftsp_square_factor_selected_power = ff_q_ftsp_square_factor_selected_power_product_factor * S ((S (ff_i_ftsp_square_factor_selected_power_product)) * ff_c_ftsp_square_factor_selected_power) + (ff_p_ftsp_square_factor_selected_power_product))) /\ ((((exists ff_h_ftsp_square_factor_selected_power_product_partial. ff_h_ftsp_square_factor_selected_power_product_partial + S (ff_r_ftsp_square_factor_selected_power_product) = S ((S (ff_i_ftsp_square_factor_selected_power_product)) * ff_v_ftsp_square_factor_selected_power_product)) /\ exists ff_q_ftsp_square_factor_selected_power_product_partial. ff_u_ftsp_square_factor_selected_power_product = ff_q_ftsp_square_factor_selected_power_product_partial * S ((S (ff_i_ftsp_square_factor_selected_power_product)) * ff_v_ftsp_square_factor_selected_power_product) + (ff_r_ftsp_square_factor_selected_power_product))) /\ ((((exists ff_h_ftsp_square_factor_selected_power_product_successor. ff_h_ftsp_square_factor_selected_power_product_successor + S (ff_s_ftsp_square_factor_selected_power_product) = S ((S (S ff_i_ftsp_square_factor_selected_power_product)) * ff_v_ftsp_square_factor_selected_power_product)) /\ exists ff_q_ftsp_square_factor_selected_power_product_successor. ff_u_ftsp_square_factor_selected_power_product = ff_q_ftsp_square_factor_selected_power_product_successor * S ((S (S ff_i_ftsp_square_factor_selected_power_product)) * ff_v_ftsp_square_factor_selected_power_product) + (ff_s_ftsp_square_factor_selected_power_product))) /\ ff_s_ftsp_square_factor_selected_power_product = ff_r_ftsp_square_factor_selected_power_product * ff_p_ftsp_square_factor_selected_power_product)))))))) /\ (exists bpv_factor_ftsp_square_factor_selected_divides. z = bpv_result_ftsp_square_factor_selected * bpv_factor_ftsp_square_factor_selected_divides)))) /\ forall bpv_candidate_ftsp_square_factor. (exists bpv_gap_ftsp_square_factor_candidate_bound. bpv_gap_ftsp_square_factor_candidate_bound + bpv_candidate_ftsp_square_factor = z) -> (exists bpv_result_ftsp_square_factor_candidate. ((exists ff_b_ftsp_square_factor_candidate_power ff_c_ftsp_square_factor_candidate_power. ((forall ff_i_ftsp_square_factor_candidate_power_repeat. (exists ff_lt_ftsp_square_factor_candidate_power_repeat_bound. ff_lt_ftsp_square_factor_candidate_power_repeat_bound + S ff_i_ftsp_square_factor_candidate_power_repeat = bpv_candidate_ftsp_square_factor) -> (((exists ff_h_ftsp_square_factor_candidate_power_repeat_decoded. ff_h_ftsp_square_factor_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsp_square_factor_candidate_power_repeat)) * ff_c_ftsp_square_factor_candidate_power)) /\ exists ff_q_ftsp_square_factor_candidate_power_repeat_decoded. ff_b_ftsp_square_factor_candidate_power = ff_q_ftsp_square_factor_candidate_power_repeat_decoded * S ((S (ff_i_ftsp_square_factor_candidate_power_repeat)) * ff_c_ftsp_square_factor_candidate_power) + (p)))) /\ (exists ff_u_ftsp_square_factor_candidate_power_product ff_v_ftsp_square_factor_candidate_power_product. ((((exists ff_h_ftsp_square_factor_candidate_power_product_start. ff_h_ftsp_square_factor_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsp_square_factor_candidate_power_product)) /\ exists ff_q_ftsp_square_factor_candidate_power_product_start. ff_u_ftsp_square_factor_candidate_power_product = ff_q_ftsp_square_factor_candidate_power_product_start * S ((S (0)) * ff_v_ftsp_square_factor_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsp_square_factor_candidate_power_product_terminal. ff_h_ftsp_square_factor_candidate_power_product_terminal + S (bpv_result_ftsp_square_factor_candidate) = S ((S (bpv_candidate_ftsp_square_factor)) * ff_v_ftsp_square_factor_candidate_power_product)) /\ exists ff_q_ftsp_square_factor_candidate_power_product_terminal. ff_u_ftsp_square_factor_candidate_power_product = ff_q_ftsp_square_factor_candidate_power_product_terminal * S ((S (bpv_candidate_ftsp_square_factor)) * ff_v_ftsp_square_factor_candidate_power_product) + (bpv_result_ftsp_square_factor_candidate))) /\ forall ff_i_ftsp_square_factor_candidate_power_product. (exists ff_lt_ftsp_square_factor_candidate_power_product_bound. ff_lt_ftsp_square_factor_candidate_power_product_bound + S ff_i_ftsp_square_factor_candidate_power_product = bpv_candidate_ftsp_square_factor) -> exists ff_p_ftsp_square_factor_candidate_power_product ff_r_ftsp_square_factor_candidate_power_product ff_s_ftsp_square_factor_candidate_power_product. ((((exists ff_h_ftsp_square_factor_candidate_power_product_factor. ff_h_ftsp_square_factor_candidate_power_product_factor + S (ff_p_ftsp_square_factor_candidate_power_product) = S ((S (ff_i_ftsp_square_factor_candidate_power_product)) * ff_c_ftsp_square_factor_candidate_power)) /\ exists ff_q_ftsp_square_factor_candidate_power_product_factor. ff_b_ftsp_square_factor_candidate_power = ff_q_ftsp_square_factor_candidate_power_product_factor * S ((S (ff_i_ftsp_square_factor_candidate_power_product)) * ff_c_ftsp_square_factor_candidate_power) + (ff_p_ftsp_square_factor_candidate_power_product))) /\ ((((exists ff_h_ftsp_square_factor_candidate_power_product_partial. ff_h_ftsp_square_factor_candidate_power_product_partial + S (ff_r_ftsp_square_factor_candidate_power_product) = S ((S (ff_i_ftsp_square_factor_candidate_power_product)) * ff_v_ftsp_square_factor_candidate_power_product)) /\ exists ff_q_ftsp_square_factor_candidate_power_product_partial. ff_u_ftsp_square_factor_candidate_power_product = ff_q_ftsp_square_factor_candidate_power_product_partial * S ((S (ff_i_ftsp_square_factor_candidate_power_product)) * ff_v_ftsp_square_factor_candidate_power_product) + (ff_r_ftsp_square_factor_candidate_power_product))) /\ ((((exists ff_h_ftsp_square_factor_candidate_power_product_successor. ff_h_ftsp_square_factor_candidate_power_product_successor + S (ff_s_ftsp_square_factor_candidate_power_product) = S ((S (S ff_i_ftsp_square_factor_candidate_power_product)) * ff_v_ftsp_square_factor_candidate_power_product)) /\ exists ff_q_ftsp_square_factor_candidate_power_product_successor. ff_u_ftsp_square_factor_candidate_power_product = ff_q_ftsp_square_factor_candidate_power_product_successor * S ((S (S ff_i_ftsp_square_factor_candidate_power_product)) * ff_v_ftsp_square_factor_candidate_power_product) + (ff_s_ftsp_square_factor_candidate_power_product))) /\ ff_s_ftsp_square_factor_candidate_power_product = ff_r_ftsp_square_factor_candidate_power_product * ff_p_ftsp_square_factor_candidate_power_product)))))))) /\ (exists bpv_factor_ftsp_square_factor_candidate_divides. z = bpv_result_ftsp_square_factor_candidate * bpv_factor_ftsp_square_factor_candidate_divides))) -> (exists bpv_gap_ftsp_square_factor_maximal. bpv_gap_ftsp_square_factor_maximal + bpv_candidate_ftsp_square_factor = e)) -> (((exists bpv_gap_ftsp_square_cofactor_exponent_bound. bpv_gap_ftsp_square_cofactor_exponent_bound + f = n) /\ (exists bpv_result_ftsp_square_cofactor_selected. ((exists ff_b_ftsp_square_cofactor_selected_power ff_c_ftsp_square_cofactor_selected_power. ((forall ff_i_ftsp_square_cofactor_selected_power_repeat. (exists ff_lt_ftsp_square_cofactor_selected_power_repeat_bound. ff_lt_ftsp_square_cofactor_selected_power_repeat_bound + S ff_i_ftsp_square_cofactor_selected_power_repeat = f) -> (((exists ff_h_ftsp_square_cofactor_selected_power_repeat_decoded. ff_h_ftsp_square_cofactor_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsp_square_cofactor_selected_power_repeat)) * ff_c_ftsp_square_cofactor_selected_power)) /\ exists ff_q_ftsp_square_cofactor_selected_power_repeat_decoded. ff_b_ftsp_square_cofactor_selected_power = ff_q_ftsp_square_cofactor_selected_power_repeat_decoded * S ((S (ff_i_ftsp_square_cofactor_selected_power_repeat)) * ff_c_ftsp_square_cofactor_selected_power) + (p)))) /\ (exists ff_u_ftsp_square_cofactor_selected_power_product ff_v_ftsp_square_cofactor_selected_power_product. ((((exists ff_h_ftsp_square_cofactor_selected_power_product_start. ff_h_ftsp_square_cofactor_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsp_square_cofactor_selected_power_product)) /\ exists ff_q_ftsp_square_cofactor_selected_power_product_start. ff_u_ftsp_square_cofactor_selected_power_product = ff_q_ftsp_square_cofactor_selected_power_product_start * S ((S (0)) * ff_v_ftsp_square_cofactor_selected_power_product) + (1))) /\ ((((exists ff_h_ftsp_square_cofactor_selected_power_product_terminal. ff_h_ftsp_square_cofactor_selected_power_product_terminal + S (bpv_result_ftsp_square_cofactor_selected) = S ((S (f)) * ff_v_ftsp_square_cofactor_selected_power_product)) /\ exists ff_q_ftsp_square_cofactor_selected_power_product_terminal. ff_u_ftsp_square_cofactor_selected_power_product = ff_q_ftsp_square_cofactor_selected_power_product_terminal * S ((S (f)) * ff_v_ftsp_square_cofactor_selected_power_product) + (bpv_result_ftsp_square_cofactor_selected))) /\ forall ff_i_ftsp_square_cofactor_selected_power_product. (exists ff_lt_ftsp_square_cofactor_selected_power_product_bound. ff_lt_ftsp_square_cofactor_selected_power_product_bound + S ff_i_ftsp_square_cofactor_selected_power_product = f) -> exists ff_p_ftsp_square_cofactor_selected_power_product ff_r_ftsp_square_cofactor_selected_power_product ff_s_ftsp_square_cofactor_selected_power_product. ((((exists ff_h_ftsp_square_cofactor_selected_power_product_factor. ff_h_ftsp_square_cofactor_selected_power_product_factor + S (ff_p_ftsp_square_cofactor_selected_power_product) = S ((S (ff_i_ftsp_square_cofactor_selected_power_product)) * ff_c_ftsp_square_cofactor_selected_power)) /\ exists ff_q_ftsp_square_cofactor_selected_power_product_factor. ff_b_ftsp_square_cofactor_selected_power = ff_q_ftsp_square_cofactor_selected_power_product_factor * S ((S (ff_i_ftsp_square_cofactor_selected_power_product)) * ff_c_ftsp_square_cofactor_selected_power) + (ff_p_ftsp_square_cofactor_selected_power_product))) /\ ((((exists ff_h_ftsp_square_cofactor_selected_power_product_partial. ff_h_ftsp_square_cofactor_selected_power_product_partial + S (ff_r_ftsp_square_cofactor_selected_power_product) = S ((S (ff_i_ftsp_square_cofactor_selected_power_product)) * ff_v_ftsp_square_cofactor_selected_power_product)) /\ exists ff_q_ftsp_square_cofactor_selected_power_product_partial. ff_u_ftsp_square_cofactor_selected_power_product = ff_q_ftsp_square_cofactor_selected_power_product_partial * S ((S (ff_i_ftsp_square_cofactor_selected_power_product)) * ff_v_ftsp_square_cofactor_selected_power_product) + (ff_r_ftsp_square_cofactor_selected_power_product))) /\ ((((exists ff_h_ftsp_square_cofactor_selected_power_product_successor. ff_h_ftsp_square_cofactor_selected_power_product_successor + S (ff_s_ftsp_square_cofactor_selected_power_product) = S ((S (S ff_i_ftsp_square_cofactor_selected_power_product)) * ff_v_ftsp_square_cofactor_selected_power_product)) /\ exists ff_q_ftsp_square_cofactor_selected_power_product_successor. ff_u_ftsp_square_cofactor_selected_power_product = ff_q_ftsp_square_cofactor_selected_power_product_successor * S ((S (S ff_i_ftsp_square_cofactor_selected_power_product)) * ff_v_ftsp_square_cofactor_selected_power_product) + (ff_s_ftsp_square_cofactor_selected_power_product))) /\ ff_s_ftsp_square_cofactor_selected_power_product = ff_r_ftsp_square_cofactor_selected_power_product * ff_p_ftsp_square_cofactor_selected_power_product)))))))) /\ (exists bpv_factor_ftsp_square_cofactor_selected_divides. n = bpv_result_ftsp_square_cofactor_selected * bpv_factor_ftsp_square_cofactor_selected_divides)))) /\ forall bpv_candidate_ftsp_square_cofactor. (exists bpv_gap_ftsp_square_cofactor_candidate_bound. bpv_gap_ftsp_square_cofactor_candidate_bound + bpv_candidate_ftsp_square_cofactor = n) -> (exists bpv_result_ftsp_square_cofactor_candidate. ((exists ff_b_ftsp_square_cofactor_candidate_power ff_c_ftsp_square_cofactor_candidate_power. ((forall ff_i_ftsp_square_cofactor_candidate_power_repeat. (exists ff_lt_ftsp_square_cofactor_candidate_power_repeat_bound. ff_lt_ftsp_square_cofactor_candidate_power_repeat_bound + S ff_i_ftsp_square_cofactor_candidate_power_repeat = bpv_candidate_ftsp_square_cofactor) -> (((exists ff_h_ftsp_square_cofactor_candidate_power_repeat_decoded. ff_h_ftsp_square_cofactor_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsp_square_cofactor_candidate_power_repeat)) * ff_c_ftsp_square_cofactor_candidate_power)) /\ exists ff_q_ftsp_square_cofactor_candidate_power_repeat_decoded. ff_b_ftsp_square_cofactor_candidate_power = ff_q_ftsp_square_cofactor_candidate_power_repeat_decoded * S ((S (ff_i_ftsp_square_cofactor_candidate_power_repeat)) * ff_c_ftsp_square_cofactor_candidate_power) + (p)))) /\ (exists ff_u_ftsp_square_cofactor_candidate_power_product ff_v_ftsp_square_cofactor_candidate_power_product. ((((exists ff_h_ftsp_square_cofactor_candidate_power_product_start. ff_h_ftsp_square_cofactor_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsp_square_cofactor_candidate_power_product)) /\ exists ff_q_ftsp_square_cofactor_candidate_power_product_start. ff_u_ftsp_square_cofactor_candidate_power_product = ff_q_ftsp_square_cofactor_candidate_power_product_start * S ((S (0)) * ff_v_ftsp_square_cofactor_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsp_square_cofactor_candidate_power_product_terminal. ff_h_ftsp_square_cofactor_candidate_power_product_terminal + S (bpv_result_ftsp_square_cofactor_candidate) = S ((S (bpv_candidate_ftsp_square_cofactor)) * ff_v_ftsp_square_cofactor_candidate_power_product)) /\ exists ff_q_ftsp_square_cofactor_candidate_power_product_terminal. ff_u_ftsp_square_cofactor_candidate_power_product = ff_q_ftsp_square_cofactor_candidate_power_product_terminal * S ((S (bpv_candidate_ftsp_square_cofactor)) * ff_v_ftsp_square_cofactor_candidate_power_product) + (bpv_result_ftsp_square_cofactor_candidate))) /\ forall ff_i_ftsp_square_cofactor_candidate_power_product. (exists ff_lt_ftsp_square_cofactor_candidate_power_product_bound. ff_lt_ftsp_square_cofactor_candidate_power_product_bound + S ff_i_ftsp_square_cofactor_candidate_power_product = bpv_candidate_ftsp_square_cofactor) -> exists ff_p_ftsp_square_cofactor_candidate_power_product ff_r_ftsp_square_cofactor_candidate_power_product ff_s_ftsp_square_cofactor_candidate_power_product. ((((exists ff_h_ftsp_square_cofactor_candidate_power_product_factor. ff_h_ftsp_square_cofactor_candidate_power_product_factor + S (ff_p_ftsp_square_cofactor_candidate_power_product) = S ((S (ff_i_ftsp_square_cofactor_candidate_power_product)) * ff_c_ftsp_square_cofactor_candidate_power)) /\ exists ff_q_ftsp_square_cofactor_candidate_power_product_factor. ff_b_ftsp_square_cofactor_candidate_power = ff_q_ftsp_square_cofactor_candidate_power_product_factor * S ((S (ff_i_ftsp_square_cofactor_candidate_power_product)) * ff_c_ftsp_square_cofactor_candidate_power) + (ff_p_ftsp_square_cofactor_candidate_power_product))) /\ ((((exists ff_h_ftsp_square_cofactor_candidate_power_product_partial. ff_h_ftsp_square_cofactor_candidate_power_product_partial + S (ff_r_ftsp_square_cofactor_candidate_power_product) = S ((S (ff_i_ftsp_square_cofactor_candidate_power_product)) * ff_v_ftsp_square_cofactor_candidate_power_product)) /\ exists ff_q_ftsp_square_cofactor_candidate_power_product_partial. ff_u_ftsp_square_cofactor_candidate_power_product = ff_q_ftsp_square_cofactor_candidate_power_product_partial * S ((S (ff_i_ftsp_square_cofactor_candidate_power_product)) * ff_v_ftsp_square_cofactor_candidate_power_product) + (ff_r_ftsp_square_cofactor_candidate_power_product))) /\ ((((exists ff_h_ftsp_square_cofactor_candidate_power_product_successor. ff_h_ftsp_square_cofactor_candidate_power_product_successor + S (ff_s_ftsp_square_cofactor_candidate_power_product) = S ((S (S ff_i_ftsp_square_cofactor_candidate_power_product)) * ff_v_ftsp_square_cofactor_candidate_power_product)) /\ exists ff_q_ftsp_square_cofactor_candidate_power_product_successor. ff_u_ftsp_square_cofactor_candidate_power_product = ff_q_ftsp_square_cofactor_candidate_power_product_successor * S ((S (S ff_i_ftsp_square_cofactor_candidate_power_product)) * ff_v_ftsp_square_cofactor_candidate_power_product) + (ff_s_ftsp_square_cofactor_candidate_power_product))) /\ ff_s_ftsp_square_cofactor_candidate_power_product = ff_r_ftsp_square_cofactor_candidate_power_product * ff_p_ftsp_square_cofactor_candidate_power_product)))))))) /\ (exists bpv_factor_ftsp_square_cofactor_candidate_divides. n = bpv_result_ftsp_square_cofactor_candidate * bpv_factor_ftsp_square_cofactor_candidate_divides))) -> (exists bpv_gap_ftsp_square_cofactor_maximal. bpv_gap_ftsp_square_cofactor_maximal + bpv_candidate_ftsp_square_cofactor = f)) -> (((exists bpv_gap_ftsp_square_product_exponent_bound. bpv_gap_ftsp_square_product_exponent_bound + g = ((z * z) * n)) /\ (exists bpv_result_ftsp_square_product_selected. ((exists ff_b_ftsp_square_product_selected_power ff_c_ftsp_square_product_selected_power. ((forall ff_i_ftsp_square_product_selected_power_repeat. (exists ff_lt_ftsp_square_product_selected_power_repeat_bound. ff_lt_ftsp_square_product_selected_power_repeat_bound + S ff_i_ftsp_square_product_selected_power_repeat = g) -> (((exists ff_h_ftsp_square_product_selected_power_repeat_decoded. ff_h_ftsp_square_product_selected_power_repeat_decoded + S (p) = S ((S (ff_i_ftsp_square_product_selected_power_repeat)) * ff_c_ftsp_square_product_selected_power)) /\ exists ff_q_ftsp_square_product_selected_power_repeat_decoded. ff_b_ftsp_square_product_selected_power = ff_q_ftsp_square_product_selected_power_repeat_decoded * S ((S (ff_i_ftsp_square_product_selected_power_repeat)) * ff_c_ftsp_square_product_selected_power) + (p)))) /\ (exists ff_u_ftsp_square_product_selected_power_product ff_v_ftsp_square_product_selected_power_product. ((((exists ff_h_ftsp_square_product_selected_power_product_start. ff_h_ftsp_square_product_selected_power_product_start + S (1) = S ((S (0)) * ff_v_ftsp_square_product_selected_power_product)) /\ exists ff_q_ftsp_square_product_selected_power_product_start. ff_u_ftsp_square_product_selected_power_product = ff_q_ftsp_square_product_selected_power_product_start * S ((S (0)) * ff_v_ftsp_square_product_selected_power_product) + (1))) /\ ((((exists ff_h_ftsp_square_product_selected_power_product_terminal. ff_h_ftsp_square_product_selected_power_product_terminal + S (bpv_result_ftsp_square_product_selected) = S ((S (g)) * ff_v_ftsp_square_product_selected_power_product)) /\ exists ff_q_ftsp_square_product_selected_power_product_terminal. ff_u_ftsp_square_product_selected_power_product = ff_q_ftsp_square_product_selected_power_product_terminal * S ((S (g)) * ff_v_ftsp_square_product_selected_power_product) + (bpv_result_ftsp_square_product_selected))) /\ forall ff_i_ftsp_square_product_selected_power_product. (exists ff_lt_ftsp_square_product_selected_power_product_bound. ff_lt_ftsp_square_product_selected_power_product_bound + S ff_i_ftsp_square_product_selected_power_product = g) -> exists ff_p_ftsp_square_product_selected_power_product ff_r_ftsp_square_product_selected_power_product ff_s_ftsp_square_product_selected_power_product. ((((exists ff_h_ftsp_square_product_selected_power_product_factor. ff_h_ftsp_square_product_selected_power_product_factor + S (ff_p_ftsp_square_product_selected_power_product) = S ((S (ff_i_ftsp_square_product_selected_power_product)) * ff_c_ftsp_square_product_selected_power)) /\ exists ff_q_ftsp_square_product_selected_power_product_factor. ff_b_ftsp_square_product_selected_power = ff_q_ftsp_square_product_selected_power_product_factor * S ((S (ff_i_ftsp_square_product_selected_power_product)) * ff_c_ftsp_square_product_selected_power) + (ff_p_ftsp_square_product_selected_power_product))) /\ ((((exists ff_h_ftsp_square_product_selected_power_product_partial. ff_h_ftsp_square_product_selected_power_product_partial + S (ff_r_ftsp_square_product_selected_power_product) = S ((S (ff_i_ftsp_square_product_selected_power_product)) * ff_v_ftsp_square_product_selected_power_product)) /\ exists ff_q_ftsp_square_product_selected_power_product_partial. ff_u_ftsp_square_product_selected_power_product = ff_q_ftsp_square_product_selected_power_product_partial * S ((S (ff_i_ftsp_square_product_selected_power_product)) * ff_v_ftsp_square_product_selected_power_product) + (ff_r_ftsp_square_product_selected_power_product))) /\ ((((exists ff_h_ftsp_square_product_selected_power_product_successor. ff_h_ftsp_square_product_selected_power_product_successor + S (ff_s_ftsp_square_product_selected_power_product) = S ((S (S ff_i_ftsp_square_product_selected_power_product)) * ff_v_ftsp_square_product_selected_power_product)) /\ exists ff_q_ftsp_square_product_selected_power_product_successor. ff_u_ftsp_square_product_selected_power_product = ff_q_ftsp_square_product_selected_power_product_successor * S ((S (S ff_i_ftsp_square_product_selected_power_product)) * ff_v_ftsp_square_product_selected_power_product) + (ff_s_ftsp_square_product_selected_power_product))) /\ ff_s_ftsp_square_product_selected_power_product = ff_r_ftsp_square_product_selected_power_product * ff_p_ftsp_square_product_selected_power_product)))))))) /\ (exists bpv_factor_ftsp_square_product_selected_divides. ((z * z) * n) = bpv_result_ftsp_square_product_selected * bpv_factor_ftsp_square_product_selected_divides)))) /\ forall bpv_candidate_ftsp_square_product. (exists bpv_gap_ftsp_square_product_candidate_bound. bpv_gap_ftsp_square_product_candidate_bound + bpv_candidate_ftsp_square_product = ((z * z) * n)) -> (exists bpv_result_ftsp_square_product_candidate. ((exists ff_b_ftsp_square_product_candidate_power ff_c_ftsp_square_product_candidate_power. ((forall ff_i_ftsp_square_product_candidate_power_repeat. (exists ff_lt_ftsp_square_product_candidate_power_repeat_bound. ff_lt_ftsp_square_product_candidate_power_repeat_bound + S ff_i_ftsp_square_product_candidate_power_repeat = bpv_candidate_ftsp_square_product) -> (((exists ff_h_ftsp_square_product_candidate_power_repeat_decoded. ff_h_ftsp_square_product_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_ftsp_square_product_candidate_power_repeat)) * ff_c_ftsp_square_product_candidate_power)) /\ exists ff_q_ftsp_square_product_candidate_power_repeat_decoded. ff_b_ftsp_square_product_candidate_power = ff_q_ftsp_square_product_candidate_power_repeat_decoded * S ((S (ff_i_ftsp_square_product_candidate_power_repeat)) * ff_c_ftsp_square_product_candidate_power) + (p)))) /\ (exists ff_u_ftsp_square_product_candidate_power_product ff_v_ftsp_square_product_candidate_power_product. ((((exists ff_h_ftsp_square_product_candidate_power_product_start. ff_h_ftsp_square_product_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_ftsp_square_product_candidate_power_product)) /\ exists ff_q_ftsp_square_product_candidate_power_product_start. ff_u_ftsp_square_product_candidate_power_product = ff_q_ftsp_square_product_candidate_power_product_start * S ((S (0)) * ff_v_ftsp_square_product_candidate_power_product) + (1))) /\ ((((exists ff_h_ftsp_square_product_candidate_power_product_terminal. ff_h_ftsp_square_product_candidate_power_product_terminal + S (bpv_result_ftsp_square_product_candidate) = S ((S (bpv_candidate_ftsp_square_product)) * ff_v_ftsp_square_product_candidate_power_product)) /\ exists ff_q_ftsp_square_product_candidate_power_product_terminal. ff_u_ftsp_square_product_candidate_power_product = ff_q_ftsp_square_product_candidate_power_product_terminal * S ((S (bpv_candidate_ftsp_square_product)) * ff_v_ftsp_square_product_candidate_power_product) + (bpv_result_ftsp_square_product_candidate))) /\ forall ff_i_ftsp_square_product_candidate_power_product. (exists ff_lt_ftsp_square_product_candidate_power_product_bound. ff_lt_ftsp_square_product_candidate_power_product_bound + S ff_i_ftsp_square_product_candidate_power_product = bpv_candidate_ftsp_square_product) -> exists ff_p_ftsp_square_product_candidate_power_product ff_r_ftsp_square_product_candidate_power_product ff_s_ftsp_square_product_candidate_power_product. ((((exists ff_h_ftsp_square_product_candidate_power_product_factor. ff_h_ftsp_square_product_candidate_power_product_factor + S (ff_p_ftsp_square_product_candidate_power_product) = S ((S (ff_i_ftsp_square_product_candidate_power_product)) * ff_c_ftsp_square_product_candidate_power)) /\ exists ff_q_ftsp_square_product_candidate_power_product_factor. ff_b_ftsp_square_product_candidate_power = ff_q_ftsp_square_product_candidate_power_product_factor * S ((S (ff_i_ftsp_square_product_candidate_power_product)) * ff_c_ftsp_square_product_candidate_power) + (ff_p_ftsp_square_product_candidate_power_product))) /\ ((((exists ff_h_ftsp_square_product_candidate_power_product_partial. ff_h_ftsp_square_product_candidate_power_product_partial + S (ff_r_ftsp_square_product_candidate_power_product) = S ((S (ff_i_ftsp_square_product_candidate_power_product)) * ff_v_ftsp_square_product_candidate_power_product)) /\ exists ff_q_ftsp_square_product_candidate_power_product_partial. ff_u_ftsp_square_product_candidate_power_product = ff_q_ftsp_square_product_candidate_power_product_partial * S ((S (ff_i_ftsp_square_product_candidate_power_product)) * ff_v_ftsp_square_product_candidate_power_product) + (ff_r_ftsp_square_product_candidate_power_product))) /\ ((((exists ff_h_ftsp_square_product_candidate_power_product_successor. ff_h_ftsp_square_product_candidate_power_product_successor + S (ff_s_ftsp_square_product_candidate_power_product) = S ((S (S ff_i_ftsp_square_product_candidate_power_product)) * ff_v_ftsp_square_product_candidate_power_product)) /\ exists ff_q_ftsp_square_product_candidate_power_product_successor. ff_u_ftsp_square_product_candidate_power_product = ff_q_ftsp_square_product_candidate_power_product_successor * S ((S (S ff_i_ftsp_square_product_candidate_power_product)) * ff_v_ftsp_square_product_candidate_power_product) + (ff_s_ftsp_square_product_candidate_power_product))) /\ ff_s_ftsp_square_product_candidate_power_product = ff_r_ftsp_square_product_candidate_power_product * ff_p_ftsp_square_product_candidate_power_product)))))))) /\ (exists bpv_factor_ftsp_square_product_candidate_divides. ((z * z) * n) = bpv_result_ftsp_square_product_candidate * bpv_factor_ftsp_square_product_candidate_divides))) -> (exists bpv_gap_ftsp_square_product_maximal. bpv_gap_ftsp_square_product_maximal + bpv_candidate_ftsp_square_product = g)) -> (exists h. g = h + h) -> exists h. f = h + h

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

In local proof propositions

Proof neighborhood

Direct theorem prerequisites

Direct theorem dependents

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.