MatrixProductPrefix(lb,lc,rb,rc,w,v,tb,tc,l)

Hygienic expanded first-order definition

forall ff_index_mcp_prefix_advanced. (exists mcp_gap_advanced_index. mcp_gap_advanced_index + S (ff_index_mcp_prefix_advanced) = (l)) -> exists ff_row_mcp_prefix_advanced ff_column_mcp_prefix_advanced ff_value_mcp_prefix_advanced. ((ff_index_mcp_prefix_advanced) = (v) * ff_row_mcp_prefix_advanced + ff_column_mcp_prefix_advanced /\ ((exists mcp_gap_advanced_column. mcp_gap_advanced_column + S (ff_column_mcp_prefix_advanced) = (v)) /\ ((exists ff_left_mcp_cell_advanced_cell ff_left_scale_mcp_cell_advanced_cell ff_right_mcp_cell_advanced_cell ff_right_scale_mcp_cell_advanced_cell. ((forall ff_index_mcp_advanced_cell_row ff_source_mcp_advanced_cell_row ff_target_mcp_advanced_cell_row. (exists mcp_gap_advanced_cell_row_bound. mcp_gap_advanced_cell_row_bound + S (ff_index_mcp_advanced_cell_row) = (w)) -> (((exists fs_h_mcp_advanced_cell_row_source. fs_h_mcp_advanced_cell_row_source + S (ff_source_mcp_advanced_cell_row) = S ((S (((ff_row_mcp_prefix_advanced) * (w)) + (1) * ff_index_mcp_advanced_cell_row)) * lc)) /\ exists fs_q_mcp_advanced_cell_row_source. lb = fs_q_mcp_advanced_cell_row_source * S ((S (((ff_row_mcp_prefix_advanced) * (w)) + (1) * ff_index_mcp_advanced_cell_row)) * lc) + (ff_source_mcp_advanced_cell_row))) -> (((exists fs_h_mcp_advanced_cell_row_target. fs_h_mcp_advanced_cell_row_target + S (ff_target_mcp_advanced_cell_row) = S ((S (ff_index_mcp_advanced_cell_row)) * ff_left_scale_mcp_cell_advanced_cell)) /\ exists fs_q_mcp_advanced_cell_row_target. ff_left_mcp_cell_advanced_cell = fs_q_mcp_advanced_cell_row_target * S ((S (ff_index_mcp_advanced_cell_row)) * ff_left_scale_mcp_cell_advanced_cell) + (ff_target_mcp_advanced_cell_row))) -> ff_target_mcp_advanced_cell_row = ff_source_mcp_advanced_cell_row) /\ ((forall ff_index_mcp_advanced_cell_column ff_source_mcp_advanced_cell_column ff_target_mcp_advanced_cell_column. (exists mcp_gap_advanced_cell_column_bound. mcp_gap_advanced_cell_column_bound + S (ff_index_mcp_advanced_cell_column) = (w)) -> (((exists fs_h_mcp_advanced_cell_column_source. fs_h_mcp_advanced_cell_column_source + S (ff_source_mcp_advanced_cell_column) = S ((S ((ff_column_mcp_prefix_advanced) + (v) * ff_index_mcp_advanced_cell_column)) * rc)) /\ exists fs_q_mcp_advanced_cell_column_source. rb = fs_q_mcp_advanced_cell_column_source * S ((S ((ff_column_mcp_prefix_advanced) + (v) * ff_index_mcp_advanced_cell_column)) * rc) + (ff_source_mcp_advanced_cell_column))) -> (((exists fs_h_mcp_advanced_cell_column_target. fs_h_mcp_advanced_cell_column_target + S (ff_target_mcp_advanced_cell_column) = S ((S (ff_index_mcp_advanced_cell_column)) * ff_right_scale_mcp_cell_advanced_cell)) /\ exists fs_q_mcp_advanced_cell_column_target. ff_right_mcp_cell_advanced_cell = fs_q_mcp_advanced_cell_column_target * S ((S (ff_index_mcp_advanced_cell_column)) * ff_right_scale_mcp_cell_advanced_cell) + (ff_target_mcp_advanced_cell_column))) -> ff_target_mcp_advanced_cell_column = ff_source_mcp_advanced_cell_column) /\ (exists ff_code_dot_mcp_advanced_cell_dot ff_scale_dot_mcp_advanced_cell_dot. ((forall fpmp_index_dot_mcp_advanced_cell_dot_pointwise fpmp_left_dot_mcp_advanced_cell_dot_pointwise fpmp_right_dot_mcp_advanced_cell_dot_pointwise fpmp_target_dot_mcp_advanced_cell_dot_pointwise. (exists fpmp_gap_dot_mcp_advanced_cell_dot_pointwise. fpmp_gap_dot_mcp_advanced_cell_dot_pointwise + S fpmp_index_dot_mcp_advanced_cell_dot_pointwise = w) -> (((exists ff_h_fpmp_dot_mcp_advanced_cell_dot_pointwise_left. ff_h_fpmp_dot_mcp_advanced_cell_dot_pointwise_left + S (fpmp_left_dot_mcp_advanced_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_advanced_cell_dot_pointwise)) * ff_left_scale_mcp_cell_advanced_cell)) /\ exists ff_q_fpmp_dot_mcp_advanced_cell_dot_pointwise_left. ff_left_mcp_cell_advanced_cell = ff_q_fpmp_dot_mcp_advanced_cell_dot_pointwise_left * S ((S (fpmp_index_dot_mcp_advanced_cell_dot_pointwise)) * ff_left_scale_mcp_cell_advanced_cell) + (fpmp_left_dot_mcp_advanced_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_advanced_cell_dot_pointwise_right. ff_h_fpmp_dot_mcp_advanced_cell_dot_pointwise_right + S (fpmp_right_dot_mcp_advanced_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_advanced_cell_dot_pointwise)) * ff_right_scale_mcp_cell_advanced_cell)) /\ exists ff_q_fpmp_dot_mcp_advanced_cell_dot_pointwise_right. ff_right_mcp_cell_advanced_cell = ff_q_fpmp_dot_mcp_advanced_cell_dot_pointwise_right * S ((S (fpmp_index_dot_mcp_advanced_cell_dot_pointwise)) * ff_right_scale_mcp_cell_advanced_cell) + (fpmp_right_dot_mcp_advanced_cell_dot_pointwise))) -> (((exists ff_h_fpmp_dot_mcp_advanced_cell_dot_pointwise_target. ff_h_fpmp_dot_mcp_advanced_cell_dot_pointwise_target + S (fpmp_target_dot_mcp_advanced_cell_dot_pointwise) = S ((S (fpmp_index_dot_mcp_advanced_cell_dot_pointwise)) * ff_scale_dot_mcp_advanced_cell_dot)) /\ exists ff_q_fpmp_dot_mcp_advanced_cell_dot_pointwise_target. ff_code_dot_mcp_advanced_cell_dot = ff_q_fpmp_dot_mcp_advanced_cell_dot_pointwise_target * S ((S (fpmp_index_dot_mcp_advanced_cell_dot_pointwise)) * ff_scale_dot_mcp_advanced_cell_dot) + (fpmp_target_dot_mcp_advanced_cell_dot_pointwise))) -> fpmp_target_dot_mcp_advanced_cell_dot_pointwise = fpmp_left_dot_mcp_advanced_cell_dot_pointwise * fpmp_right_dot_mcp_advanced_cell_dot_pointwise) /\ (exists ff_u_dot_mcp_advanced_cell_dot_sum ff_v_dot_mcp_advanced_cell_dot_sum. ((((exists ff_h_dot_mcp_advanced_cell_dot_sum_start. ff_h_dot_mcp_advanced_cell_dot_sum_start + S (0) = S ((S (0)) * ff_v_dot_mcp_advanced_cell_dot_sum)) /\ exists ff_q_dot_mcp_advanced_cell_dot_sum_start. ff_u_dot_mcp_advanced_cell_dot_sum = ff_q_dot_mcp_advanced_cell_dot_sum_start * S ((S (0)) * ff_v_dot_mcp_advanced_cell_dot_sum) + (0))) /\ ((((exists ff_h_dot_mcp_advanced_cell_dot_sum_terminal. ff_h_dot_mcp_advanced_cell_dot_sum_terminal + S (ff_value_mcp_prefix_advanced) = S ((S (w)) * ff_v_dot_mcp_advanced_cell_dot_sum)) /\ exists ff_q_dot_mcp_advanced_cell_dot_sum_terminal. ff_u_dot_mcp_advanced_cell_dot_sum = ff_q_dot_mcp_advanced_cell_dot_sum_terminal * S ((S (w)) * ff_v_dot_mcp_advanced_cell_dot_sum) + (ff_value_mcp_prefix_advanced))) /\ forall ff_i_dot_mcp_advanced_cell_dot_sum. (exists ff_lt_dot_mcp_advanced_cell_dot_sum_bound. ff_lt_dot_mcp_advanced_cell_dot_sum_bound + S ff_i_dot_mcp_advanced_cell_dot_sum = w) -> exists ff_a_dot_mcp_advanced_cell_dot_sum ff_r_dot_mcp_advanced_cell_dot_sum ff_s_dot_mcp_advanced_cell_dot_sum. ((((exists ff_h_dot_mcp_advanced_cell_dot_sum_summand. ff_h_dot_mcp_advanced_cell_dot_sum_summand + S (ff_a_dot_mcp_advanced_cell_dot_sum) = S ((S (ff_i_dot_mcp_advanced_cell_dot_sum)) * ff_scale_dot_mcp_advanced_cell_dot)) /\ exists ff_q_dot_mcp_advanced_cell_dot_sum_summand. ff_code_dot_mcp_advanced_cell_dot = ff_q_dot_mcp_advanced_cell_dot_sum_summand * S ((S (ff_i_dot_mcp_advanced_cell_dot_sum)) * ff_scale_dot_mcp_advanced_cell_dot) + (ff_a_dot_mcp_advanced_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_advanced_cell_dot_sum_partial. ff_h_dot_mcp_advanced_cell_dot_sum_partial + S (ff_r_dot_mcp_advanced_cell_dot_sum) = S ((S (ff_i_dot_mcp_advanced_cell_dot_sum)) * ff_v_dot_mcp_advanced_cell_dot_sum)) /\ exists ff_q_dot_mcp_advanced_cell_dot_sum_partial. ff_u_dot_mcp_advanced_cell_dot_sum = ff_q_dot_mcp_advanced_cell_dot_sum_partial * S ((S (ff_i_dot_mcp_advanced_cell_dot_sum)) * ff_v_dot_mcp_advanced_cell_dot_sum) + (ff_r_dot_mcp_advanced_cell_dot_sum))) /\ ((((exists ff_h_dot_mcp_advanced_cell_dot_sum_successor. ff_h_dot_mcp_advanced_cell_dot_sum_successor + S (ff_s_dot_mcp_advanced_cell_dot_sum) = S ((S (S ff_i_dot_mcp_advanced_cell_dot_sum)) * ff_v_dot_mcp_advanced_cell_dot_sum)) /\ exists ff_q_dot_mcp_advanced_cell_dot_sum_successor. ff_u_dot_mcp_advanced_cell_dot_sum = ff_q_dot_mcp_advanced_cell_dot_sum_successor * S ((S (S ff_i_dot_mcp_advanced_cell_dot_sum)) * ff_v_dot_mcp_advanced_cell_dot_sum) + (ff_s_dot_mcp_advanced_cell_dot_sum))) /\ ff_s_dot_mcp_advanced_cell_dot_sum = ff_r_dot_mcp_advanced_cell_dot_sum + ff_a_dot_mcp_advanced_cell_dot_sum))))))))))) /\ (((exists fs_h_mcp_advanced_entry. fs_h_mcp_advanced_entry + S (ff_value_mcp_prefix_advanced) = S ((S (ff_index_mcp_prefix_advanced)) * tc)) /\ exists fs_q_mcp_advanced_entry. tb = fs_q_mcp_advanced_entry * S ((S (ff_index_mcp_prefix_advanced)) * tc) + (ff_value_mcp_prefix_advanced))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Definitions depending on this notation

Checked theorems using this definition