choose_functional

Exact expanded PA statement

forall n k x y. (((exists bcf_lt_gap_bclf_left_out_of_range. bcf_lt_gap_bclf_left_out_of_range + S (n) = k) /\ x = 0) \/ ((exists bcf_le_gap_bclf_left_in_range. bcf_le_gap_bclf_left_in_range + (k) = n) /\ (exists bcf_row_code_code_bclf_left bcf_row_code_scale_bclf_left bcf_row_scale_code_bclf_left bcf_row_scale_scale_bclf_left bcf_row_code_bclf_left bcf_row_scale_bclf_left. ((forall bcf_row_index_bclf_left_table. (exists bcf_lt_gap_bclf_left_table_row_bound. bcf_lt_gap_bclf_left_table_row_bound + S (bcf_row_index_bclf_left_table) = S (n)) -> exists bcf_row_code_bclf_left_table bcf_row_scale_bclf_left_table. ((((exists bcf_height_bclf_left_table_decoded_row_code. bcf_height_bclf_left_table_decoded_row_code + S (bcf_row_code_bclf_left_table) = S ((S (bcf_row_index_bclf_left_table)) * bcf_row_code_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_table_decoded_row_code. bcf_row_code_code_bclf_left = bcf_quotient_bclf_left_table_decoded_row_code * S ((S (bcf_row_index_bclf_left_table)) * bcf_row_code_scale_bclf_left) + (bcf_row_code_bclf_left_table))) /\ ((((exists bcf_height_bclf_left_table_decoded_row_scale. bcf_height_bclf_left_table_decoded_row_scale + S (bcf_row_scale_bclf_left_table) = S ((S (bcf_row_index_bclf_left_table)) * bcf_row_scale_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_table_decoded_row_scale. bcf_row_scale_code_bclf_left = bcf_quotient_bclf_left_table_decoded_row_scale * S ((S (bcf_row_index_bclf_left_table)) * bcf_row_scale_scale_bclf_left) + (bcf_row_scale_bclf_left_table))) /\ ((bcf_row_index_bclf_left_table = 0 /\ (forall bcf_index_bclf_left_table_zero_row. (exists bcf_lt_gap_bclf_left_table_zero_row_bound. bcf_lt_gap_bclf_left_table_zero_row_bound + S (bcf_index_bclf_left_table_zero_row) = S (n)) -> exists bcf_value_bclf_left_table_zero_row. ((((exists bcf_height_bclf_left_table_zero_row_entry. bcf_height_bclf_left_table_zero_row_entry + S (bcf_value_bclf_left_table_zero_row) = S ((S (bcf_index_bclf_left_table_zero_row)) * bcf_row_scale_bclf_left_table)) /\ exists bcf_quotient_bclf_left_table_zero_row_entry. bcf_row_code_bclf_left_table = bcf_quotient_bclf_left_table_zero_row_entry * S ((S (bcf_index_bclf_left_table_zero_row)) * bcf_row_scale_bclf_left_table) + (bcf_value_bclf_left_table_zero_row))) /\ ((bcf_index_bclf_left_table_zero_row = 0 /\ bcf_value_bclf_left_table_zero_row = 1) \/ exists bcf_predecessor_bclf_left_table_zero_row. bcf_index_bclf_left_table_zero_row = S bcf_predecessor_bclf_left_table_zero_row /\ bcf_value_bclf_left_table_zero_row = 0)))) \/ exists bcf_predecessor_bclf_left_table bcf_previous_code_bclf_left_table bcf_previous_scale_bclf_left_table. bcf_row_index_bclf_left_table = S bcf_predecessor_bclf_left_table /\ ((((exists bcf_height_bclf_left_table_decoded_previous_code. bcf_height_bclf_left_table_decoded_previous_code + S (bcf_previous_code_bclf_left_table) = S ((S (bcf_predecessor_bclf_left_table)) * bcf_row_code_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_table_decoded_previous_code. bcf_row_code_code_bclf_left = bcf_quotient_bclf_left_table_decoded_previous_code * S ((S (bcf_predecessor_bclf_left_table)) * bcf_row_code_scale_bclf_left) + (bcf_previous_code_bclf_left_table))) /\ ((((exists bcf_height_bclf_left_table_decoded_previous_scale. bcf_height_bclf_left_table_decoded_previous_scale + S (bcf_previous_scale_bclf_left_table) = S ((S (bcf_predecessor_bclf_left_table)) * bcf_row_scale_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_table_decoded_previous_scale. bcf_row_scale_code_bclf_left = bcf_quotient_bclf_left_table_decoded_previous_scale * S ((S (bcf_predecessor_bclf_left_table)) * bcf_row_scale_scale_bclf_left) + (bcf_previous_scale_bclf_left_table))) /\ (forall bcf_index_bclf_left_table_row_step. (exists bcf_lt_gap_bclf_left_table_row_step_bound. bcf_lt_gap_bclf_left_table_row_step_bound + S (bcf_index_bclf_left_table_row_step) = S (n)) -> exists bcf_value_bclf_left_table_row_step. ((((exists bcf_height_bclf_left_table_row_step_entry. bcf_height_bclf_left_table_row_step_entry + S (bcf_value_bclf_left_table_row_step) = S ((S (bcf_index_bclf_left_table_row_step)) * bcf_row_scale_bclf_left_table)) /\ exists bcf_quotient_bclf_left_table_row_step_entry. bcf_row_code_bclf_left_table = bcf_quotient_bclf_left_table_row_step_entry * S ((S (bcf_index_bclf_left_table_row_step)) * bcf_row_scale_bclf_left_table) + (bcf_value_bclf_left_table_row_step))) /\ ((bcf_index_bclf_left_table_row_step = 0 /\ bcf_value_bclf_left_table_row_step = 1) \/ exists bcf_predecessor_bclf_left_table_row_step bcf_left_bclf_left_table_row_step bcf_right_bclf_left_table_row_step. bcf_index_bclf_left_table_row_step = S bcf_predecessor_bclf_left_table_row_step /\ ((((exists bcf_height_bclf_left_table_row_step_previous_left. bcf_height_bclf_left_table_row_step_previous_left + S (bcf_left_bclf_left_table_row_step) = S ((S (bcf_predecessor_bclf_left_table_row_step)) * bcf_previous_scale_bclf_left_table)) /\ exists bcf_quotient_bclf_left_table_row_step_previous_left. bcf_previous_code_bclf_left_table = bcf_quotient_bclf_left_table_row_step_previous_left * S ((S (bcf_predecessor_bclf_left_table_row_step)) * bcf_previous_scale_bclf_left_table) + (bcf_left_bclf_left_table_row_step))) /\ ((((exists bcf_height_bclf_left_table_row_step_previous_right. bcf_height_bclf_left_table_row_step_previous_right + S (bcf_right_bclf_left_table_row_step) = S ((S (S (bcf_predecessor_bclf_left_table_row_step))) * bcf_previous_scale_bclf_left_table)) /\ exists bcf_quotient_bclf_left_table_row_step_previous_right. bcf_previous_code_bclf_left_table = bcf_quotient_bclf_left_table_row_step_previous_right * S ((S (S (bcf_predecessor_bclf_left_table_row_step))) * bcf_previous_scale_bclf_left_table) + (bcf_right_bclf_left_table_row_step))) /\ bcf_value_bclf_left_table_row_step = bcf_left_bclf_left_table_row_step + bcf_right_bclf_left_table_row_step))))))))))) /\ ((((exists bcf_height_bclf_left_decoded_row_code. bcf_height_bclf_left_decoded_row_code + S (bcf_row_code_bclf_left) = S ((S (n)) * bcf_row_code_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_decoded_row_code. bcf_row_code_code_bclf_left = bcf_quotient_bclf_left_decoded_row_code * S ((S (n)) * bcf_row_code_scale_bclf_left) + (bcf_row_code_bclf_left))) /\ ((((exists bcf_height_bclf_left_decoded_row_scale. bcf_height_bclf_left_decoded_row_scale + S (bcf_row_scale_bclf_left) = S ((S (n)) * bcf_row_scale_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_decoded_row_scale. bcf_row_scale_code_bclf_left = bcf_quotient_bclf_left_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_bclf_left) + (bcf_row_scale_bclf_left))) /\ (((exists bcf_height_bclf_left_decoded_value. bcf_height_bclf_left_decoded_value + S (x) = S ((S (k)) * bcf_row_scale_bclf_left)) /\ exists bcf_quotient_bclf_left_decoded_value. bcf_row_code_bclf_left = bcf_quotient_bclf_left_decoded_value * S ((S (k)) * bcf_row_scale_bclf_left) + (x))))))))) -> (((exists bcf_lt_gap_bclf_right_out_of_range. bcf_lt_gap_bclf_right_out_of_range + S (n) = k) /\ y = 0) \/ ((exists bcf_le_gap_bclf_right_in_range. bcf_le_gap_bclf_right_in_range + (k) = n) /\ (exists bcf_row_code_code_bclf_right bcf_row_code_scale_bclf_right bcf_row_scale_code_bclf_right bcf_row_scale_scale_bclf_right bcf_row_code_bclf_right bcf_row_scale_bclf_right. ((forall bcf_row_index_bclf_right_table. (exists bcf_lt_gap_bclf_right_table_row_bound. bcf_lt_gap_bclf_right_table_row_bound + S (bcf_row_index_bclf_right_table) = S (n)) -> exists bcf_row_code_bclf_right_table bcf_row_scale_bclf_right_table. ((((exists bcf_height_bclf_right_table_decoded_row_code. bcf_height_bclf_right_table_decoded_row_code + S (bcf_row_code_bclf_right_table) = S ((S (bcf_row_index_bclf_right_table)) * bcf_row_code_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_table_decoded_row_code. bcf_row_code_code_bclf_right = bcf_quotient_bclf_right_table_decoded_row_code * S ((S (bcf_row_index_bclf_right_table)) * bcf_row_code_scale_bclf_right) + (bcf_row_code_bclf_right_table))) /\ ((((exists bcf_height_bclf_right_table_decoded_row_scale. bcf_height_bclf_right_table_decoded_row_scale + S (bcf_row_scale_bclf_right_table) = S ((S (bcf_row_index_bclf_right_table)) * bcf_row_scale_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_table_decoded_row_scale. bcf_row_scale_code_bclf_right = bcf_quotient_bclf_right_table_decoded_row_scale * S ((S (bcf_row_index_bclf_right_table)) * bcf_row_scale_scale_bclf_right) + (bcf_row_scale_bclf_right_table))) /\ ((bcf_row_index_bclf_right_table = 0 /\ (forall bcf_index_bclf_right_table_zero_row. (exists bcf_lt_gap_bclf_right_table_zero_row_bound. bcf_lt_gap_bclf_right_table_zero_row_bound + S (bcf_index_bclf_right_table_zero_row) = S (n)) -> exists bcf_value_bclf_right_table_zero_row. ((((exists bcf_height_bclf_right_table_zero_row_entry. bcf_height_bclf_right_table_zero_row_entry + S (bcf_value_bclf_right_table_zero_row) = S ((S (bcf_index_bclf_right_table_zero_row)) * bcf_row_scale_bclf_right_table)) /\ exists bcf_quotient_bclf_right_table_zero_row_entry. bcf_row_code_bclf_right_table = bcf_quotient_bclf_right_table_zero_row_entry * S ((S (bcf_index_bclf_right_table_zero_row)) * bcf_row_scale_bclf_right_table) + (bcf_value_bclf_right_table_zero_row))) /\ ((bcf_index_bclf_right_table_zero_row = 0 /\ bcf_value_bclf_right_table_zero_row = 1) \/ exists bcf_predecessor_bclf_right_table_zero_row. bcf_index_bclf_right_table_zero_row = S bcf_predecessor_bclf_right_table_zero_row /\ bcf_value_bclf_right_table_zero_row = 0)))) \/ exists bcf_predecessor_bclf_right_table bcf_previous_code_bclf_right_table bcf_previous_scale_bclf_right_table. bcf_row_index_bclf_right_table = S bcf_predecessor_bclf_right_table /\ ((((exists bcf_height_bclf_right_table_decoded_previous_code. bcf_height_bclf_right_table_decoded_previous_code + S (bcf_previous_code_bclf_right_table) = S ((S (bcf_predecessor_bclf_right_table)) * bcf_row_code_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_table_decoded_previous_code. bcf_row_code_code_bclf_right = bcf_quotient_bclf_right_table_decoded_previous_code * S ((S (bcf_predecessor_bclf_right_table)) * bcf_row_code_scale_bclf_right) + (bcf_previous_code_bclf_right_table))) /\ ((((exists bcf_height_bclf_right_table_decoded_previous_scale. bcf_height_bclf_right_table_decoded_previous_scale + S (bcf_previous_scale_bclf_right_table) = S ((S (bcf_predecessor_bclf_right_table)) * bcf_row_scale_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_table_decoded_previous_scale. bcf_row_scale_code_bclf_right = bcf_quotient_bclf_right_table_decoded_previous_scale * S ((S (bcf_predecessor_bclf_right_table)) * bcf_row_scale_scale_bclf_right) + (bcf_previous_scale_bclf_right_table))) /\ (forall bcf_index_bclf_right_table_row_step. (exists bcf_lt_gap_bclf_right_table_row_step_bound. bcf_lt_gap_bclf_right_table_row_step_bound + S (bcf_index_bclf_right_table_row_step) = S (n)) -> exists bcf_value_bclf_right_table_row_step. ((((exists bcf_height_bclf_right_table_row_step_entry. bcf_height_bclf_right_table_row_step_entry + S (bcf_value_bclf_right_table_row_step) = S ((S (bcf_index_bclf_right_table_row_step)) * bcf_row_scale_bclf_right_table)) /\ exists bcf_quotient_bclf_right_table_row_step_entry. bcf_row_code_bclf_right_table = bcf_quotient_bclf_right_table_row_step_entry * S ((S (bcf_index_bclf_right_table_row_step)) * bcf_row_scale_bclf_right_table) + (bcf_value_bclf_right_table_row_step))) /\ ((bcf_index_bclf_right_table_row_step = 0 /\ bcf_value_bclf_right_table_row_step = 1) \/ exists bcf_predecessor_bclf_right_table_row_step bcf_left_bclf_right_table_row_step bcf_right_bclf_right_table_row_step. bcf_index_bclf_right_table_row_step = S bcf_predecessor_bclf_right_table_row_step /\ ((((exists bcf_height_bclf_right_table_row_step_previous_left. bcf_height_bclf_right_table_row_step_previous_left + S (bcf_left_bclf_right_table_row_step) = S ((S (bcf_predecessor_bclf_right_table_row_step)) * bcf_previous_scale_bclf_right_table)) /\ exists bcf_quotient_bclf_right_table_row_step_previous_left. bcf_previous_code_bclf_right_table = bcf_quotient_bclf_right_table_row_step_previous_left * S ((S (bcf_predecessor_bclf_right_table_row_step)) * bcf_previous_scale_bclf_right_table) + (bcf_left_bclf_right_table_row_step))) /\ ((((exists bcf_height_bclf_right_table_row_step_previous_right. bcf_height_bclf_right_table_row_step_previous_right + S (bcf_right_bclf_right_table_row_step) = S ((S (S (bcf_predecessor_bclf_right_table_row_step))) * bcf_previous_scale_bclf_right_table)) /\ exists bcf_quotient_bclf_right_table_row_step_previous_right. bcf_previous_code_bclf_right_table = bcf_quotient_bclf_right_table_row_step_previous_right * S ((S (S (bcf_predecessor_bclf_right_table_row_step))) * bcf_previous_scale_bclf_right_table) + (bcf_right_bclf_right_table_row_step))) /\ bcf_value_bclf_right_table_row_step = bcf_left_bclf_right_table_row_step + bcf_right_bclf_right_table_row_step))))))))))) /\ ((((exists bcf_height_bclf_right_decoded_row_code. bcf_height_bclf_right_decoded_row_code + S (bcf_row_code_bclf_right) = S ((S (n)) * bcf_row_code_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_decoded_row_code. bcf_row_code_code_bclf_right = bcf_quotient_bclf_right_decoded_row_code * S ((S (n)) * bcf_row_code_scale_bclf_right) + (bcf_row_code_bclf_right))) /\ ((((exists bcf_height_bclf_right_decoded_row_scale. bcf_height_bclf_right_decoded_row_scale + S (bcf_row_scale_bclf_right) = S ((S (n)) * bcf_row_scale_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_decoded_row_scale. bcf_row_scale_code_bclf_right = bcf_quotient_bclf_right_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_bclf_right) + (bcf_row_scale_bclf_right))) /\ (((exists bcf_height_bclf_right_decoded_value. bcf_height_bclf_right_decoded_value + S (y) = S ((S (k)) * bcf_row_scale_bclf_right)) /\ exists bcf_quotient_bclf_right_decoded_value. bcf_row_code_bclf_right = bcf_quotient_bclf_right_decoded_value * S ((S (k)) * bcf_row_scale_bclf_right) + (y))))))))) -> x = y