power_divides_add_mul

Exact expanded PA statement

forall p e f s a b. s = e + f -> (exists bpv_result_add_mul_left. ((exists ff_b_add_mul_left_power ff_c_add_mul_left_power. ((forall ff_i_add_mul_left_power_repeat. (exists ff_lt_add_mul_left_power_repeat_bound. ff_lt_add_mul_left_power_repeat_bound + S ff_i_add_mul_left_power_repeat = e) -> (((exists ff_h_add_mul_left_power_repeat_decoded. ff_h_add_mul_left_power_repeat_decoded + S (p) = S ((S (ff_i_add_mul_left_power_repeat)) * ff_c_add_mul_left_power)) /\ exists ff_q_add_mul_left_power_repeat_decoded. ff_b_add_mul_left_power = ff_q_add_mul_left_power_repeat_decoded * S ((S (ff_i_add_mul_left_power_repeat)) * ff_c_add_mul_left_power) + (p)))) /\ (exists ff_u_add_mul_left_power_product ff_v_add_mul_left_power_product. ((((exists ff_h_add_mul_left_power_product_start. ff_h_add_mul_left_power_product_start + S (1) = S ((S (0)) * ff_v_add_mul_left_power_product)) /\ exists ff_q_add_mul_left_power_product_start. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_start * S ((S (0)) * ff_v_add_mul_left_power_product) + (1))) /\ ((((exists ff_h_add_mul_left_power_product_terminal. ff_h_add_mul_left_power_product_terminal + S (bpv_result_add_mul_left) = S ((S (e)) * ff_v_add_mul_left_power_product)) /\ exists ff_q_add_mul_left_power_product_terminal. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_terminal * S ((S (e)) * ff_v_add_mul_left_power_product) + (bpv_result_add_mul_left))) /\ forall ff_i_add_mul_left_power_product. (exists ff_lt_add_mul_left_power_product_bound. ff_lt_add_mul_left_power_product_bound + S ff_i_add_mul_left_power_product = e) -> exists ff_p_add_mul_left_power_product ff_r_add_mul_left_power_product ff_s_add_mul_left_power_product. ((((exists ff_h_add_mul_left_power_product_factor. ff_h_add_mul_left_power_product_factor + S (ff_p_add_mul_left_power_product) = S ((S (ff_i_add_mul_left_power_product)) * ff_c_add_mul_left_power)) /\ exists ff_q_add_mul_left_power_product_factor. ff_b_add_mul_left_power = ff_q_add_mul_left_power_product_factor * S ((S (ff_i_add_mul_left_power_product)) * ff_c_add_mul_left_power) + (ff_p_add_mul_left_power_product))) /\ ((((exists ff_h_add_mul_left_power_product_partial. ff_h_add_mul_left_power_product_partial + S (ff_r_add_mul_left_power_product) = S ((S (ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product)) /\ exists ff_q_add_mul_left_power_product_partial. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_partial * S ((S (ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product) + (ff_r_add_mul_left_power_product))) /\ ((((exists ff_h_add_mul_left_power_product_successor. ff_h_add_mul_left_power_product_successor + S (ff_s_add_mul_left_power_product) = S ((S (S ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product)) /\ exists ff_q_add_mul_left_power_product_successor. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_successor * S ((S (S ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product) + (ff_s_add_mul_left_power_product))) /\ ff_s_add_mul_left_power_product = ff_r_add_mul_left_power_product * ff_p_add_mul_left_power_product)))))))) /\ (exists bpv_factor_add_mul_left_divides. a = bpv_result_add_mul_left * bpv_factor_add_mul_left_divides))) -> (exists bpv_result_add_mul_right. ((exists ff_b_add_mul_right_power ff_c_add_mul_right_power. ((forall ff_i_add_mul_right_power_repeat. (exists ff_lt_add_mul_right_power_repeat_bound. ff_lt_add_mul_right_power_repeat_bound + S ff_i_add_mul_right_power_repeat = f) -> (((exists ff_h_add_mul_right_power_repeat_decoded. ff_h_add_mul_right_power_repeat_decoded + S (p) = S ((S (ff_i_add_mul_right_power_repeat)) * ff_c_add_mul_right_power)) /\ exists ff_q_add_mul_right_power_repeat_decoded. ff_b_add_mul_right_power = ff_q_add_mul_right_power_repeat_decoded * S ((S (ff_i_add_mul_right_power_repeat)) * ff_c_add_mul_right_power) + (p)))) /\ (exists ff_u_add_mul_right_power_product ff_v_add_mul_right_power_product. ((((exists ff_h_add_mul_right_power_product_start. ff_h_add_mul_right_power_product_start + S (1) = S ((S (0)) * ff_v_add_mul_right_power_product)) /\ exists ff_q_add_mul_right_power_product_start. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_start * S ((S (0)) * ff_v_add_mul_right_power_product) + (1))) /\ ((((exists ff_h_add_mul_right_power_product_terminal. ff_h_add_mul_right_power_product_terminal + S (bpv_result_add_mul_right) = S ((S (f)) * ff_v_add_mul_right_power_product)) /\ exists ff_q_add_mul_right_power_product_terminal. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_terminal * S ((S (f)) * ff_v_add_mul_right_power_product) + (bpv_result_add_mul_right))) /\ forall ff_i_add_mul_right_power_product. (exists ff_lt_add_mul_right_power_product_bound. ff_lt_add_mul_right_power_product_bound + S ff_i_add_mul_right_power_product = f) -> exists ff_p_add_mul_right_power_product ff_r_add_mul_right_power_product ff_s_add_mul_right_power_product. ((((exists ff_h_add_mul_right_power_product_factor. ff_h_add_mul_right_power_product_factor + S (ff_p_add_mul_right_power_product) = S ((S (ff_i_add_mul_right_power_product)) * ff_c_add_mul_right_power)) /\ exists ff_q_add_mul_right_power_product_factor. ff_b_add_mul_right_power = ff_q_add_mul_right_power_product_factor * S ((S (ff_i_add_mul_right_power_product)) * ff_c_add_mul_right_power) + (ff_p_add_mul_right_power_product))) /\ ((((exists ff_h_add_mul_right_power_product_partial. ff_h_add_mul_right_power_product_partial + S (ff_r_add_mul_right_power_product) = S ((S (ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product)) /\ exists ff_q_add_mul_right_power_product_partial. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_partial * S ((S (ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product) + (ff_r_add_mul_right_power_product))) /\ ((((exists ff_h_add_mul_right_power_product_successor. ff_h_add_mul_right_power_product_successor + S (ff_s_add_mul_right_power_product) = S ((S (S ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product)) /\ exists ff_q_add_mul_right_power_product_successor. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_successor * S ((S (S ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product) + (ff_s_add_mul_right_power_product))) /\ ff_s_add_mul_right_power_product = ff_r_add_mul_right_power_product * ff_p_add_mul_right_power_product)))))))) /\ (exists bpv_factor_add_mul_right_divides. b = bpv_result_add_mul_right * bpv_factor_add_mul_right_divides))) -> (exists bpvi_result_add_mul_result. ((exists bpvi_b_add_mul_result_power bpvi_c_add_mul_result_power. ((forall bpvi_i_add_mul_result_power. (exists bpvi_repeat_gap_add_mul_result_power. bpvi_repeat_gap_add_mul_result_power + S bpvi_i_add_mul_result_power = s) -> (((exists bpvi_h_add_mul_result_power_repeat. bpvi_h_add_mul_result_power_repeat + S (p) = S ((S (bpvi_i_add_mul_result_power)) * bpvi_c_add_mul_result_power)) /\ exists bpvi_q_add_mul_result_power_repeat. bpvi_b_add_mul_result_power = bpvi_q_add_mul_result_power_repeat * S ((S (bpvi_i_add_mul_result_power)) * bpvi_c_add_mul_result_power) + (p)))) /\ (exists bpvi_u_add_mul_result_power bpvi_v_add_mul_result_power. ((((exists bpvi_h_add_mul_result_power_start. bpvi_h_add_mul_result_power_start + S (1) = S ((S (0)) * bpvi_v_add_mul_result_power)) /\ exists bpvi_q_add_mul_result_power_start. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_start * S ((S (0)) * bpvi_v_add_mul_result_power) + (1))) /\ ((((exists bpvi_h_add_mul_result_power_terminal. bpvi_h_add_mul_result_power_terminal + S (bpvi_result_add_mul_result) = S ((S (s)) * bpvi_v_add_mul_result_power)) /\ exists bpvi_q_add_mul_result_power_terminal. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_terminal * S ((S (s)) * bpvi_v_add_mul_result_power) + (bpvi_result_add_mul_result))) /\ forall bpvi_j_add_mul_result_power. (exists bpvi_product_gap_add_mul_result_power. bpvi_product_gap_add_mul_result_power + S bpvi_j_add_mul_result_power = s) -> exists bpvi_factor_add_mul_result_power bpvi_partial_add_mul_result_power bpvi_successor_add_mul_result_power. ((((exists bpvi_h_add_mul_result_power_factor. bpvi_h_add_mul_result_power_factor + S (bpvi_factor_add_mul_result_power) = S ((S (bpvi_j_add_mul_result_power)) * bpvi_c_add_mul_result_power)) /\ exists bpvi_q_add_mul_result_power_factor. bpvi_b_add_mul_result_power = bpvi_q_add_mul_result_power_factor * S ((S (bpvi_j_add_mul_result_power)) * bpvi_c_add_mul_result_power) + (bpvi_factor_add_mul_result_power))) /\ ((((exists bpvi_h_add_mul_result_power_partial. bpvi_h_add_mul_result_power_partial + S (bpvi_partial_add_mul_result_power) = S ((S (bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power)) /\ exists bpvi_q_add_mul_result_power_partial. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_partial * S ((S (bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power) + (bpvi_partial_add_mul_result_power))) /\ ((((exists bpvi_h_add_mul_result_power_successor. bpvi_h_add_mul_result_power_successor + S (bpvi_successor_add_mul_result_power) = S ((S (S bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power)) /\ exists bpvi_q_add_mul_result_power_successor. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_successor * S ((S (S bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power) + (bpvi_successor_add_mul_result_power))) /\ bpvi_successor_add_mul_result_power = bpvi_partial_add_mul_result_power * bpvi_factor_add_mul_result_power)))))))) /\ exists bpvi_divisor_factor_add_mul_result. a * b = bpvi_result_add_mul_result * bpvi_divisor_factor_add_mul_result))