arbitrary_euler_criterion_complete

Exact expanded PA statement

forall p a n h A. p = S n -> ((~(p = 1) /\ forall frm_prime_left_eca_prime frm_prime_right_eca_prime. p = frm_prime_left_eca_prime * frm_prime_right_eca_prime -> frm_prime_left_eca_prime = 1 \/ frm_prime_right_eca_prime = 1)) -> (~(exists frm_factor_eca_not_divisor. a = p * frm_factor_eca_not_divisor)) -> n = h + h -> (exists ff_b_eca_power_endpoint ff_c_eca_power_endpoint. ((forall ff_i_eca_power_endpoint_repeat. (exists ff_lt_eca_power_endpoint_repeat_bound. ff_lt_eca_power_endpoint_repeat_bound + S ff_i_eca_power_endpoint_repeat = h) -> (((exists ff_h_eca_power_endpoint_repeat_decoded. ff_h_eca_power_endpoint_repeat_decoded + S (a) = S ((S (ff_i_eca_power_endpoint_repeat)) * ff_c_eca_power_endpoint)) /\ exists ff_q_eca_power_endpoint_repeat_decoded. ff_b_eca_power_endpoint = ff_q_eca_power_endpoint_repeat_decoded * S ((S (ff_i_eca_power_endpoint_repeat)) * ff_c_eca_power_endpoint) + (a)))) /\ (exists ff_u_eca_power_endpoint_product ff_v_eca_power_endpoint_product. ((((exists ff_h_eca_power_endpoint_product_start. ff_h_eca_power_endpoint_product_start + S (1) = S ((S (0)) * ff_v_eca_power_endpoint_product)) /\ exists ff_q_eca_power_endpoint_product_start. ff_u_eca_power_endpoint_product = ff_q_eca_power_endpoint_product_start * S ((S (0)) * ff_v_eca_power_endpoint_product) + (1))) /\ ((((exists ff_h_eca_power_endpoint_product_terminal. ff_h_eca_power_endpoint_product_terminal + S (A) = S ((S (h)) * ff_v_eca_power_endpoint_product)) /\ exists ff_q_eca_power_endpoint_product_terminal. ff_u_eca_power_endpoint_product = ff_q_eca_power_endpoint_product_terminal * S ((S (h)) * ff_v_eca_power_endpoint_product) + (A))) /\ forall ff_i_eca_power_endpoint_product. (exists ff_lt_eca_power_endpoint_product_bound. ff_lt_eca_power_endpoint_product_bound + S ff_i_eca_power_endpoint_product = h) -> exists ff_p_eca_power_endpoint_product ff_r_eca_power_endpoint_product ff_s_eca_power_endpoint_product. ((((exists ff_h_eca_power_endpoint_product_factor. ff_h_eca_power_endpoint_product_factor + S (ff_p_eca_power_endpoint_product) = S ((S (ff_i_eca_power_endpoint_product)) * ff_c_eca_power_endpoint)) /\ exists ff_q_eca_power_endpoint_product_factor. ff_b_eca_power_endpoint = ff_q_eca_power_endpoint_product_factor * S ((S (ff_i_eca_power_endpoint_product)) * ff_c_eca_power_endpoint) + (ff_p_eca_power_endpoint_product))) /\ ((((exists ff_h_eca_power_endpoint_product_partial. ff_h_eca_power_endpoint_product_partial + S (ff_r_eca_power_endpoint_product) = S ((S (ff_i_eca_power_endpoint_product)) * ff_v_eca_power_endpoint_product)) /\ exists ff_q_eca_power_endpoint_product_partial. ff_u_eca_power_endpoint_product = ff_q_eca_power_endpoint_product_partial * S ((S (ff_i_eca_power_endpoint_product)) * ff_v_eca_power_endpoint_product) + (ff_r_eca_power_endpoint_product))) /\ ((((exists ff_h_eca_power_endpoint_product_successor. ff_h_eca_power_endpoint_product_successor + S (ff_s_eca_power_endpoint_product) = S ((S (S ff_i_eca_power_endpoint_product)) * ff_v_eca_power_endpoint_product)) /\ exists ff_q_eca_power_endpoint_product_successor. ff_u_eca_power_endpoint_product = ff_q_eca_power_endpoint_product_successor * S ((S (S ff_i_eca_power_endpoint_product)) * ff_v_eca_power_endpoint_product) + (ff_s_eca_power_endpoint_product))) /\ ff_s_eca_power_endpoint_product = ff_r_eca_power_endpoint_product * ff_p_eca_power_endpoint_product)))))))) -> (((((exists qr_x_eca_qres_a. exists qr_u_eca_qres_a qr_v_eca_qres_a. qr_x_eca_qres_a * qr_x_eca_qres_a + p * qr_u_eca_qres_a = a + p * qr_v_eca_qres_a) -> (exists wpp_mod_left_eca_mod_one wpp_mod_right_eca_mod_one. (A) + p * wpp_mod_left_eca_mod_one = (1) + p * wpp_mod_right_eca_mod_one)) /\ ((exists wpp_mod_left_eca_mod_one wpp_mod_right_eca_mod_one. (A) + p * wpp_mod_left_eca_mod_one = (1) + p * wpp_mod_right_eca_mod_one) -> (exists qr_x_eca_qres_a. exists qr_u_eca_qres_a qr_v_eca_qres_a. qr_x_eca_qres_a * qr_x_eca_qres_a + p * qr_u_eca_qres_a = a + p * qr_v_eca_qres_a)))) /\ (((~(exists qr_x_eca_qres_a. exists qr_u_eca_qres_a qr_v_eca_qres_a. qr_x_eca_qres_a * qr_x_eca_qres_a + p * qr_u_eca_qres_a = a + p * qr_v_eca_qres_a) -> (exists wpp_mod_left_eca_mod_predecessor wpp_mod_right_eca_mod_predecessor. (A) + p * wpp_mod_left_eca_mod_predecessor = (n) + p * wpp_mod_right_eca_mod_predecessor)) /\ ((exists wpp_mod_left_eca_mod_predecessor wpp_mod_right_eca_mod_predecessor. (A) + p * wpp_mod_left_eca_mod_predecessor = (n) + p * wpp_mod_right_eca_mod_predecessor) -> ~(exists qr_x_eca_qres_a. exists qr_u_eca_qres_a qr_v_eca_qres_a. qr_x_eca_qres_a * qr_x_eca_qres_a + p * qr_u_eca_qres_a = a + p * qr_v_eca_qres_a)))))