beta_pointwise_mul_product_exists

Exact expanded PA statement

forall mb mc sb sc l M Sprod. (exists ff_u_recode_package_left_product ff_v_recode_package_left_product. ((((exists ff_h_recode_package_left_product_start. ff_h_recode_package_left_product_start + S (1) = S ((S (0)) * ff_v_recode_package_left_product)) /\ exists ff_q_recode_package_left_product_start. ff_u_recode_package_left_product = ff_q_recode_package_left_product_start * S ((S (0)) * ff_v_recode_package_left_product) + (1))) /\ ((((exists ff_h_recode_package_left_product_terminal. ff_h_recode_package_left_product_terminal + S (M) = S ((S (l)) * ff_v_recode_package_left_product)) /\ exists ff_q_recode_package_left_product_terminal. ff_u_recode_package_left_product = ff_q_recode_package_left_product_terminal * S ((S (l)) * ff_v_recode_package_left_product) + (M))) /\ forall ff_i_recode_package_left_product. (exists ff_lt_recode_package_left_product_bound. ff_lt_recode_package_left_product_bound + S ff_i_recode_package_left_product = l) -> exists ff_p_recode_package_left_product ff_r_recode_package_left_product ff_s_recode_package_left_product. ((((exists ff_h_recode_package_left_product_factor. ff_h_recode_package_left_product_factor + S (ff_p_recode_package_left_product) = S ((S (ff_i_recode_package_left_product)) * mc)) /\ exists ff_q_recode_package_left_product_factor. mb = ff_q_recode_package_left_product_factor * S ((S (ff_i_recode_package_left_product)) * mc) + (ff_p_recode_package_left_product))) /\ ((((exists ff_h_recode_package_left_product_partial. ff_h_recode_package_left_product_partial + S (ff_r_recode_package_left_product) = S ((S (ff_i_recode_package_left_product)) * ff_v_recode_package_left_product)) /\ exists ff_q_recode_package_left_product_partial. ff_u_recode_package_left_product = ff_q_recode_package_left_product_partial * S ((S (ff_i_recode_package_left_product)) * ff_v_recode_package_left_product) + (ff_r_recode_package_left_product))) /\ ((((exists ff_h_recode_package_left_product_successor. ff_h_recode_package_left_product_successor + S (ff_s_recode_package_left_product) = S ((S (S ff_i_recode_package_left_product)) * ff_v_recode_package_left_product)) /\ exists ff_q_recode_package_left_product_successor. ff_u_recode_package_left_product = ff_q_recode_package_left_product_successor * S ((S (S ff_i_recode_package_left_product)) * ff_v_recode_package_left_product) + (ff_s_recode_package_left_product))) /\ ff_s_recode_package_left_product = ff_r_recode_package_left_product * ff_p_recode_package_left_product)))))) -> (exists ff_u_recode_package_right_product ff_v_recode_package_right_product. ((((exists ff_h_recode_package_right_product_start. ff_h_recode_package_right_product_start + S (1) = S ((S (0)) * ff_v_recode_package_right_product)) /\ exists ff_q_recode_package_right_product_start. ff_u_recode_package_right_product = ff_q_recode_package_right_product_start * S ((S (0)) * ff_v_recode_package_right_product) + (1))) /\ ((((exists ff_h_recode_package_right_product_terminal. ff_h_recode_package_right_product_terminal + S (Sprod) = S ((S (l)) * ff_v_recode_package_right_product)) /\ exists ff_q_recode_package_right_product_terminal. ff_u_recode_package_right_product = ff_q_recode_package_right_product_terminal * S ((S (l)) * ff_v_recode_package_right_product) + (Sprod))) /\ forall ff_i_recode_package_right_product. (exists ff_lt_recode_package_right_product_bound. ff_lt_recode_package_right_product_bound + S ff_i_recode_package_right_product = l) -> exists ff_p_recode_package_right_product ff_r_recode_package_right_product ff_s_recode_package_right_product. ((((exists ff_h_recode_package_right_product_factor. ff_h_recode_package_right_product_factor + S (ff_p_recode_package_right_product) = S ((S (ff_i_recode_package_right_product)) * sc)) /\ exists ff_q_recode_package_right_product_factor. sb = ff_q_recode_package_right_product_factor * S ((S (ff_i_recode_package_right_product)) * sc) + (ff_p_recode_package_right_product))) /\ ((((exists ff_h_recode_package_right_product_partial. ff_h_recode_package_right_product_partial + S (ff_r_recode_package_right_product) = S ((S (ff_i_recode_package_right_product)) * ff_v_recode_package_right_product)) /\ exists ff_q_recode_package_right_product_partial. ff_u_recode_package_right_product = ff_q_recode_package_right_product_partial * S ((S (ff_i_recode_package_right_product)) * ff_v_recode_package_right_product) + (ff_r_recode_package_right_product))) /\ ((((exists ff_h_recode_package_right_product_successor. ff_h_recode_package_right_product_successor + S (ff_s_recode_package_right_product) = S ((S (S ff_i_recode_package_right_product)) * ff_v_recode_package_right_product)) /\ exists ff_q_recode_package_right_product_successor. ff_u_recode_package_right_product = ff_q_recode_package_right_product_successor * S ((S (S ff_i_recode_package_right_product)) * ff_v_recode_package_right_product) + (ff_s_recode_package_right_product))) /\ ff_s_recode_package_right_product = ff_r_recode_package_right_product * ff_p_recode_package_right_product)))))) -> (exists tb tc T. ((forall fpmp_index_recode_package_alignment fpmp_left_recode_package_alignment fpmp_right_recode_package_alignment fpmp_target_recode_package_alignment. (exists fpmp_gap_recode_package_alignment. fpmp_gap_recode_package_alignment + S fpmp_index_recode_package_alignment = l) -> (((exists ff_h_fpmp_recode_package_alignment_left. ff_h_fpmp_recode_package_alignment_left + S (fpmp_left_recode_package_alignment) = S ((S (fpmp_index_recode_package_alignment)) * mc)) /\ exists ff_q_fpmp_recode_package_alignment_left. mb = ff_q_fpmp_recode_package_alignment_left * S ((S (fpmp_index_recode_package_alignment)) * mc) + (fpmp_left_recode_package_alignment))) -> (((exists ff_h_fpmp_recode_package_alignment_right. ff_h_fpmp_recode_package_alignment_right + S (fpmp_right_recode_package_alignment) = S ((S (fpmp_index_recode_package_alignment)) * sc)) /\ exists ff_q_fpmp_recode_package_alignment_right. sb = ff_q_fpmp_recode_package_alignment_right * S ((S (fpmp_index_recode_package_alignment)) * sc) + (fpmp_right_recode_package_alignment))) -> (((exists ff_h_fpmp_recode_package_alignment_target. ff_h_fpmp_recode_package_alignment_target + S (fpmp_target_recode_package_alignment) = S ((S (fpmp_index_recode_package_alignment)) * tc)) /\ exists ff_q_fpmp_recode_package_alignment_target. tb = ff_q_fpmp_recode_package_alignment_target * S ((S (fpmp_index_recode_package_alignment)) * tc) + (fpmp_target_recode_package_alignment))) -> fpmp_target_recode_package_alignment = fpmp_left_recode_package_alignment * fpmp_right_recode_package_alignment) /\ ((exists ff_u_recode_package_target_product ff_v_recode_package_target_product. ((((exists ff_h_recode_package_target_product_start. ff_h_recode_package_target_product_start + S (1) = S ((S (0)) * ff_v_recode_package_target_product)) /\ exists ff_q_recode_package_target_product_start. ff_u_recode_package_target_product = ff_q_recode_package_target_product_start * S ((S (0)) * ff_v_recode_package_target_product) + (1))) /\ ((((exists ff_h_recode_package_target_product_terminal. ff_h_recode_package_target_product_terminal + S (T) = S ((S (l)) * ff_v_recode_package_target_product)) /\ exists ff_q_recode_package_target_product_terminal. ff_u_recode_package_target_product = ff_q_recode_package_target_product_terminal * S ((S (l)) * ff_v_recode_package_target_product) + (T))) /\ forall ff_i_recode_package_target_product. (exists ff_lt_recode_package_target_product_bound. ff_lt_recode_package_target_product_bound + S ff_i_recode_package_target_product = l) -> exists ff_p_recode_package_target_product ff_r_recode_package_target_product ff_s_recode_package_target_product. ((((exists ff_h_recode_package_target_product_factor. ff_h_recode_package_target_product_factor + S (ff_p_recode_package_target_product) = S ((S (ff_i_recode_package_target_product)) * tc)) /\ exists ff_q_recode_package_target_product_factor. tb = ff_q_recode_package_target_product_factor * S ((S (ff_i_recode_package_target_product)) * tc) + (ff_p_recode_package_target_product))) /\ ((((exists ff_h_recode_package_target_product_partial. ff_h_recode_package_target_product_partial + S (ff_r_recode_package_target_product) = S ((S (ff_i_recode_package_target_product)) * ff_v_recode_package_target_product)) /\ exists ff_q_recode_package_target_product_partial. ff_u_recode_package_target_product = ff_q_recode_package_target_product_partial * S ((S (ff_i_recode_package_target_product)) * ff_v_recode_package_target_product) + (ff_r_recode_package_target_product))) /\ ((((exists ff_h_recode_package_target_product_successor. ff_h_recode_package_target_product_successor + S (ff_s_recode_package_target_product) = S ((S (S ff_i_recode_package_target_product)) * ff_v_recode_package_target_product)) /\ exists ff_q_recode_package_target_product_successor. ff_u_recode_package_target_product = ff_q_recode_package_target_product_successor * S ((S (S ff_i_recode_package_target_product)) * ff_v_recode_package_target_product) + (ff_s_recode_package_target_product))) /\ ff_s_recode_package_target_product = ff_r_recode_package_target_product * ff_p_recode_package_target_product)))))) /\ T = M * Sprod)))