Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Readable signature
Choose(n, k, z)Exact expansion
((exists bcf_lt_gap_bertrand_defined_choose_out_of_range. bcf_lt_gap_bertrand_defined_choose_out_of_range + S (n) = k) /\ z = 0) \/ ((exists bcf_le_gap_bertrand_defined_choose_in_range. bcf_le_gap_bertrand_defined_choose_in_range + (k) = n) /\ (exists bcf_row_code_code_bertrand_defined_choose bcf_row_code_scale_bertrand_defined_choose bcf_row_scale_code_bertrand_defined_choose bcf_row_scale_scale_bertrand_defined_choose bcf_row_code_bertrand_defined_choose bcf_row_scale_bertrand_defined_choose. ((forall bcf_row_index_bertrand_defined_choose_table. (exists bcf_lt_gap_bertrand_defined_choose_table_row_bound. bcf_lt_gap_bertrand_defined_choose_table_row_bound + S (bcf_row_index_bertrand_defined_choose_table) = S (n)) -> exists bcf_row_code_bertrand_defined_choose_table bcf_row_scale_bertrand_defined_choose_table. ((((exists bcf_height_bertrand_defined_choose_table_decoded_row_code. bcf_height_bertrand_defined_choose_table_decoded_row_code + S (bcf_row_code_bertrand_defined_choose_table) = S ((S (bcf_row_index_bertrand_defined_choose_table)) * bcf_row_code_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_table_decoded_row_code. bcf_row_code_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_table_decoded_row_code * S ((S (bcf_row_index_bertrand_defined_choose_table)) * bcf_row_code_scale_bertrand_defined_choose) + (bcf_row_code_bertrand_defined_choose_table))) /\ ((((exists bcf_height_bertrand_defined_choose_table_decoded_row_scale. bcf_height_bertrand_defined_choose_table_decoded_row_scale + S (bcf_row_scale_bertrand_defined_choose_table) = S ((S (bcf_row_index_bertrand_defined_choose_table)) * bcf_row_scale_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_table_decoded_row_scale. bcf_row_scale_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_table_decoded_row_scale * S ((S (bcf_row_index_bertrand_defined_choose_table)) * bcf_row_scale_scale_bertrand_defined_choose) + (bcf_row_scale_bertrand_defined_choose_table))) /\ ((bcf_row_index_bertrand_defined_choose_table = 0 /\ (forall bcf_index_bertrand_defined_choose_table_zero_row. (exists bcf_lt_gap_bertrand_defined_choose_table_zero_row_bound. bcf_lt_gap_bertrand_defined_choose_table_zero_row_bound + S (bcf_index_bertrand_defined_choose_table_zero_row) = S (n)) -> exists bcf_value_bertrand_defined_choose_table_zero_row. ((((exists bcf_height_bertrand_defined_choose_table_zero_row_entry. bcf_height_bertrand_defined_choose_table_zero_row_entry + S (bcf_value_bertrand_defined_choose_table_zero_row) = S ((S (bcf_index_bertrand_defined_choose_table_zero_row)) * bcf_row_scale_bertrand_defined_choose_table)) /\ exists bcf_quotient_bertrand_defined_choose_table_zero_row_entry. bcf_row_code_bertrand_defined_choose_table = bcf_quotient_bertrand_defined_choose_table_zero_row_entry * S ((S (bcf_index_bertrand_defined_choose_table_zero_row)) * bcf_row_scale_bertrand_defined_choose_table) + (bcf_value_bertrand_defined_choose_table_zero_row))) /\ ((bcf_index_bertrand_defined_choose_table_zero_row = 0 /\ bcf_value_bertrand_defined_choose_table_zero_row = 1) \/ exists bcf_predecessor_bertrand_defined_choose_table_zero_row. bcf_index_bertrand_defined_choose_table_zero_row = S bcf_predecessor_bertrand_defined_choose_table_zero_row /\ bcf_value_bertrand_defined_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_bertrand_defined_choose_table bcf_previous_code_bertrand_defined_choose_table bcf_previous_scale_bertrand_defined_choose_table. bcf_row_index_bertrand_defined_choose_table = S bcf_predecessor_bertrand_defined_choose_table /\ ((((exists bcf_height_bertrand_defined_choose_table_decoded_previous_code. bcf_height_bertrand_defined_choose_table_decoded_previous_code + S (bcf_previous_code_bertrand_defined_choose_table) = S ((S (bcf_predecessor_bertrand_defined_choose_table)) * bcf_row_code_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_table_decoded_previous_code. bcf_row_code_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_table_decoded_previous_code * S ((S (bcf_predecessor_bertrand_defined_choose_table)) * bcf_row_code_scale_bertrand_defined_choose) + (bcf_previous_code_bertrand_defined_choose_table))) /\ ((((exists bcf_height_bertrand_defined_choose_table_decoded_previous_scale. bcf_height_bertrand_defined_choose_table_decoded_previous_scale + S (bcf_previous_scale_bertrand_defined_choose_table) = S ((S (bcf_predecessor_bertrand_defined_choose_table)) * bcf_row_scale_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_table_decoded_previous_scale. bcf_row_scale_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_bertrand_defined_choose_table)) * bcf_row_scale_scale_bertrand_defined_choose) + (bcf_previous_scale_bertrand_defined_choose_table))) /\ (forall bcf_index_bertrand_defined_choose_table_row_step. (exists bcf_lt_gap_bertrand_defined_choose_table_row_step_bound. bcf_lt_gap_bertrand_defined_choose_table_row_step_bound + S (bcf_index_bertrand_defined_choose_table_row_step) = S (n)) -> exists bcf_value_bertrand_defined_choose_table_row_step. ((((exists bcf_height_bertrand_defined_choose_table_row_step_entry. bcf_height_bertrand_defined_choose_table_row_step_entry + S (bcf_value_bertrand_defined_choose_table_row_step) = S ((S (bcf_index_bertrand_defined_choose_table_row_step)) * bcf_row_scale_bertrand_defined_choose_table)) /\ exists bcf_quotient_bertrand_defined_choose_table_row_step_entry. bcf_row_code_bertrand_defined_choose_table = bcf_quotient_bertrand_defined_choose_table_row_step_entry * S ((S (bcf_index_bertrand_defined_choose_table_row_step)) * bcf_row_scale_bertrand_defined_choose_table) + (bcf_value_bertrand_defined_choose_table_row_step))) /\ ((bcf_index_bertrand_defined_choose_table_row_step = 0 /\ bcf_value_bertrand_defined_choose_table_row_step = 1) \/ exists bcf_predecessor_bertrand_defined_choose_table_row_step bcf_left_bertrand_defined_choose_table_row_step bcf_right_bertrand_defined_choose_table_row_step. bcf_index_bertrand_defined_choose_table_row_step = S bcf_predecessor_bertrand_defined_choose_table_row_step /\ ((((exists bcf_height_bertrand_defined_choose_table_row_step_previous_left. bcf_height_bertrand_defined_choose_table_row_step_previous_left + S (bcf_left_bertrand_defined_choose_table_row_step) = S ((S (bcf_predecessor_bertrand_defined_choose_table_row_step)) * bcf_previous_scale_bertrand_defined_choose_table)) /\ exists bcf_quotient_bertrand_defined_choose_table_row_step_previous_left. bcf_previous_code_bertrand_defined_choose_table = bcf_quotient_bertrand_defined_choose_table_row_step_previous_left * S ((S (bcf_predecessor_bertrand_defined_choose_table_row_step)) * bcf_previous_scale_bertrand_defined_choose_table) + (bcf_left_bertrand_defined_choose_table_row_step))) /\ ((((exists bcf_height_bertrand_defined_choose_table_row_step_previous_right. bcf_height_bertrand_defined_choose_table_row_step_previous_right + S (bcf_right_bertrand_defined_choose_table_row_step) = S ((S (S (bcf_predecessor_bertrand_defined_choose_table_row_step))) * bcf_previous_scale_bertrand_defined_choose_table)) /\ exists bcf_quotient_bertrand_defined_choose_table_row_step_previous_right. bcf_previous_code_bertrand_defined_choose_table = bcf_quotient_bertrand_defined_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_bertrand_defined_choose_table_row_step))) * bcf_previous_scale_bertrand_defined_choose_table) + (bcf_right_bertrand_defined_choose_table_row_step))) /\ bcf_value_bertrand_defined_choose_table_row_step = bcf_left_bertrand_defined_choose_table_row_step + bcf_right_bertrand_defined_choose_table_row_step))))))))))) /\ ((((exists bcf_height_bertrand_defined_choose_decoded_row_code. bcf_height_bertrand_defined_choose_decoded_row_code + S (bcf_row_code_bertrand_defined_choose) = S ((S (n)) * bcf_row_code_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_decoded_row_code. bcf_row_code_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_decoded_row_code * S ((S (n)) * bcf_row_code_scale_bertrand_defined_choose) + (bcf_row_code_bertrand_defined_choose))) /\ ((((exists bcf_height_bertrand_defined_choose_decoded_row_scale. bcf_height_bertrand_defined_choose_decoded_row_scale + S (bcf_row_scale_bertrand_defined_choose) = S ((S (n)) * bcf_row_scale_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_decoded_row_scale. bcf_row_scale_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_bertrand_defined_choose) + (bcf_row_scale_bertrand_defined_choose))) /\ (((exists bcf_height_bertrand_defined_choose_decoded_value. bcf_height_bertrand_defined_choose_decoded_value + S (z) = S ((S (k)) * bcf_row_scale_bertrand_defined_choose)) /\ exists bcf_quotient_bertrand_defined_choose_decoded_value. bcf_row_code_bertrand_defined_choose = bcf_quotient_bertrand_defined_choose_decoded_value * S ((S (k)) * bcf_row_scale_bertrand_defined_choose) + (z))))))))This node is conservative notation, not a theorem, axiom, predicate constant, or kernel rule. Its expansion remains in the unchanged first-order language.
Definition neighborhood
Depends on conservative definitions
Used by conservative definitions
none
All transitive conservative prerequisites
Used by theorem statements or local proof propositions
LU0000 lucas_digit_carry_implies_prime_divides LU0001 lucas_prime_row_interior_divisible LU0002 lucas_choose_prime_divisor_bound LU0003 lucas_digit_carry_iff_prime_divides LU0004 lucas_digit_no_carry_iff_not_divides LU000G lucas_prime_row_initial_coefficient_one LU000H lucas_prime_row_terminal_coefficient_one LU000I lucas_prime_row_sparse_complete LU000K lucas_positive_lower_quotient_digit_coefficient_zero LU000L lucas_zero_upper_quotient_high_column_vanishes LU000M lucas_prime_block_digit_congruence LU000N lucas_one_step_division_congruence LU000P lucas_choose_zero_index_is_one LU000Q lucas_choose_zero_upper_positive_is_zero LU000T lucas_prime_row_interior_zero_mod LU000U lucas_pascal_congruence_step LU000W lucas_prime_shift_below_base LU000Z lucas_prime_shift_high_column LU0012 lucas_repeated_prime_shift_below_base LU0014 lucas_low_digit_product_congruence LU001E lucas_choose_prefix_empty LU001F lucas_choose_prefix_extend LU001G lucas_choose_prefix_exists LU001H lucas_choose_prefix_point LU001I lucas_multidigit_congruence_from_one_step LU001J lucas_terminating_multidigit_theorem_from_one_step LU001O lucas_multidigit_congruence LU001P lucas_terminating_multidigit_theorem LU001Q lucas_theorem_for_length LU001R lucas_theoremGrand-campaign planning vocabulary
Locate Binom in the global campaign vocabulary →
Reviewed Choose corresponds to blueprint Binom with checked argument positions [0, 1, 2].
The global atlas describes planning vocabulary and does not itself certify a definition or theorem. The reviewed expansion and conservative dependency DAG on this page are the actual family-local reading definitions.