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.
Statement with defined notation
∀ p. ∀ C0. Choose(p,0,C0) → C0 = 1Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.
Definitions used by this theorem
In the theorem statement
In local proof propositions
none
Exact expanded first-order statement
forall p C0. (((exists bcf_lt_gap_lucas_row_initial_out_of_range. bcf_lt_gap_lucas_row_initial_out_of_range + S (p) = 0) /\ C0 = 0) \/ ((exists bcf_le_gap_lucas_row_initial_in_range. bcf_le_gap_lucas_row_initial_in_range + (0) = p) /\ (exists bcf_row_code_code_lucas_row_initial bcf_row_code_scale_lucas_row_initial bcf_row_scale_code_lucas_row_initial bcf_row_scale_scale_lucas_row_initial bcf_row_code_lucas_row_initial bcf_row_scale_lucas_row_initial. ((forall bcf_row_index_lucas_row_initial_table. (exists bcf_lt_gap_lucas_row_initial_table_row_bound. bcf_lt_gap_lucas_row_initial_table_row_bound + S (bcf_row_index_lucas_row_initial_table) = S (p)) -> exists bcf_row_code_lucas_row_initial_table bcf_row_scale_lucas_row_initial_table. ((((exists bcf_height_lucas_row_initial_table_decoded_row_code. bcf_height_lucas_row_initial_table_decoded_row_code + S (bcf_row_code_lucas_row_initial_table) = S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_row_code. bcf_row_code_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_row_code * S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial) + (bcf_row_code_lucas_row_initial_table))) /\ ((((exists bcf_height_lucas_row_initial_table_decoded_row_scale. bcf_height_lucas_row_initial_table_decoded_row_scale + S (bcf_row_scale_lucas_row_initial_table) = S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_row_scale. bcf_row_scale_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_row_scale * S ((S (bcf_row_index_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial) + (bcf_row_scale_lucas_row_initial_table))) /\ ((bcf_row_index_lucas_row_initial_table = 0 /\ (forall bcf_index_lucas_row_initial_table_zero_row. (exists bcf_lt_gap_lucas_row_initial_table_zero_row_bound. bcf_lt_gap_lucas_row_initial_table_zero_row_bound + S (bcf_index_lucas_row_initial_table_zero_row) = S (p)) -> exists bcf_value_lucas_row_initial_table_zero_row. ((((exists bcf_height_lucas_row_initial_table_zero_row_entry. bcf_height_lucas_row_initial_table_zero_row_entry + S (bcf_value_lucas_row_initial_table_zero_row) = S ((S (bcf_index_lucas_row_initial_table_zero_row)) * bcf_row_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_zero_row_entry. bcf_row_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_zero_row_entry * S ((S (bcf_index_lucas_row_initial_table_zero_row)) * bcf_row_scale_lucas_row_initial_table) + (bcf_value_lucas_row_initial_table_zero_row))) /\ ((bcf_index_lucas_row_initial_table_zero_row = 0 /\ bcf_value_lucas_row_initial_table_zero_row = 1) \/ exists bcf_predecessor_lucas_row_initial_table_zero_row. bcf_index_lucas_row_initial_table_zero_row = S bcf_predecessor_lucas_row_initial_table_zero_row /\ bcf_value_lucas_row_initial_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_row_initial_table bcf_previous_code_lucas_row_initial_table bcf_previous_scale_lucas_row_initial_table. bcf_row_index_lucas_row_initial_table = S bcf_predecessor_lucas_row_initial_table /\ ((((exists bcf_height_lucas_row_initial_table_decoded_previous_code. bcf_height_lucas_row_initial_table_decoded_previous_code + S (bcf_previous_code_lucas_row_initial_table) = S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_previous_code. bcf_row_code_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_code_scale_lucas_row_initial) + (bcf_previous_code_lucas_row_initial_table))) /\ ((((exists bcf_height_lucas_row_initial_table_decoded_previous_scale. bcf_height_lucas_row_initial_table_decoded_previous_scale + S (bcf_previous_scale_lucas_row_initial_table) = S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_table_decoded_previous_scale. bcf_row_scale_code_lucas_row_initial = bcf_quotient_lucas_row_initial_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_row_initial_table)) * bcf_row_scale_scale_lucas_row_initial) + (bcf_previous_scale_lucas_row_initial_table))) /\ (forall bcf_index_lucas_row_initial_table_row_step. (exists bcf_lt_gap_lucas_row_initial_table_row_step_bound. bcf_lt_gap_lucas_row_initial_table_row_step_bound + S (bcf_index_lucas_row_initial_table_row_step) = S (p)) -> exists bcf_value_lucas_row_initial_table_row_step. ((((exists bcf_height_lucas_row_initial_table_row_step_entry. bcf_height_lucas_row_initial_table_row_step_entry + S (bcf_value_lucas_row_initial_table_row_step) = S ((S (bcf_index_lucas_row_initial_table_row_step)) * bcf_row_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_row_step_entry. bcf_row_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_row_step_entry * S ((S (bcf_index_lucas_row_initial_table_row_step)) * bcf_row_scale_lucas_row_initial_table) + (bcf_value_lucas_row_initial_table_row_step))) /\ ((bcf_index_lucas_row_initial_table_row_step = 0 /\ bcf_value_lucas_row_initial_table_row_step = 1) \/ exists bcf_predecessor_lucas_row_initial_table_row_step bcf_left_lucas_row_initial_table_row_step bcf_right_lucas_row_initial_table_row_step. bcf_index_lucas_row_initial_table_row_step = S bcf_predecessor_lucas_row_initial_table_row_step /\ ((((exists bcf_height_lucas_row_initial_table_row_step_previous_left. bcf_height_lucas_row_initial_table_row_step_previous_left + S (bcf_left_lucas_row_initial_table_row_step) = S ((S (bcf_predecessor_lucas_row_initial_table_row_step)) * bcf_previous_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_row_step_previous_left. bcf_previous_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_row_initial_table_row_step)) * bcf_previous_scale_lucas_row_initial_table) + (bcf_left_lucas_row_initial_table_row_step))) /\ ((((exists bcf_height_lucas_row_initial_table_row_step_previous_right. bcf_height_lucas_row_initial_table_row_step_previous_right + S (bcf_right_lucas_row_initial_table_row_step) = S ((S (S (bcf_predecessor_lucas_row_initial_table_row_step))) * bcf_previous_scale_lucas_row_initial_table)) /\ exists bcf_quotient_lucas_row_initial_table_row_step_previous_right. bcf_previous_code_lucas_row_initial_table = bcf_quotient_lucas_row_initial_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_row_initial_table_row_step))) * bcf_previous_scale_lucas_row_initial_table) + (bcf_right_lucas_row_initial_table_row_step))) /\ bcf_value_lucas_row_initial_table_row_step = bcf_left_lucas_row_initial_table_row_step + bcf_right_lucas_row_initial_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_row_initial_decoded_row_code. bcf_height_lucas_row_initial_decoded_row_code + S (bcf_row_code_lucas_row_initial) = S ((S (p)) * bcf_row_code_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_decoded_row_code. bcf_row_code_code_lucas_row_initial = bcf_quotient_lucas_row_initial_decoded_row_code * S ((S (p)) * bcf_row_code_scale_lucas_row_initial) + (bcf_row_code_lucas_row_initial))) /\ ((((exists bcf_height_lucas_row_initial_decoded_row_scale. bcf_height_lucas_row_initial_decoded_row_scale + S (bcf_row_scale_lucas_row_initial) = S ((S (p)) * bcf_row_scale_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_decoded_row_scale. bcf_row_scale_code_lucas_row_initial = bcf_quotient_lucas_row_initial_decoded_row_scale * S ((S (p)) * bcf_row_scale_scale_lucas_row_initial) + (bcf_row_scale_lucas_row_initial))) /\ (((exists bcf_height_lucas_row_initial_decoded_value. bcf_height_lucas_row_initial_decoded_value + S (C0) = S ((S (0)) * bcf_row_scale_lucas_row_initial)) /\ exists bcf_quotient_lucas_row_initial_decoded_value. bcf_row_code_lucas_row_initial = bcf_quotient_lucas_row_initial_decoded_value * S ((S (0)) * bcf_row_scale_lucas_row_initial) + (C0))))))))) -> C0 = S (0)Proof neighborhood
Direct theorem prerequisites
choose_zero · Alpha closed
Direct theorem dependents
Definition-aware tactic body
Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.
Read the argument
Proof checkpoints
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.