LU000P

lucas_choose_zero_index_is_one

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every relational binomial coefficient with a witnessed zero lower index is exactly one.

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.

Exact expanded first-order arithmetic statement

forall n k C. k = 0 -> (((exists bcf_lt_gap_lucas_convolution_zero_index_source_out_of_range. bcf_lt_gap_lucas_convolution_zero_index_source_out_of_range + S (n) = k) /\ C = 0) \/ ((exists bcf_le_gap_lucas_convolution_zero_index_source_in_range. bcf_le_gap_lucas_convolution_zero_index_source_in_range + (k) = n) /\ (exists bcf_row_code_code_lucas_convolution_zero_index_source bcf_row_code_scale_lucas_convolution_zero_index_source bcf_row_scale_code_lucas_convolution_zero_index_source bcf_row_scale_scale_lucas_convolution_zero_index_source bcf_row_code_lucas_convolution_zero_index_source bcf_row_scale_lucas_convolution_zero_index_source. ((forall bcf_row_index_lucas_convolution_zero_index_source_table. (exists bcf_lt_gap_lucas_convolution_zero_index_source_table_row_bound. bcf_lt_gap_lucas_convolution_zero_index_source_table_row_bound + S (bcf_row_index_lucas_convolution_zero_index_source_table) = S (n)) -> exists bcf_row_code_lucas_convolution_zero_index_source_table bcf_row_scale_lucas_convolution_zero_index_source_table. ((((exists bcf_height_lucas_convolution_zero_index_source_table_decoded_row_code. bcf_height_lucas_convolution_zero_index_source_table_decoded_row_code + S (bcf_row_code_lucas_convolution_zero_index_source_table) = S ((S (bcf_row_index_lucas_convolution_zero_index_source_table)) * bcf_row_code_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_decoded_row_code. bcf_row_code_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_table_decoded_row_code * S ((S (bcf_row_index_lucas_convolution_zero_index_source_table)) * bcf_row_code_scale_lucas_convolution_zero_index_source) + (bcf_row_code_lucas_convolution_zero_index_source_table))) /\ ((((exists bcf_height_lucas_convolution_zero_index_source_table_decoded_row_scale. bcf_height_lucas_convolution_zero_index_source_table_decoded_row_scale + S (bcf_row_scale_lucas_convolution_zero_index_source_table) = S ((S (bcf_row_index_lucas_convolution_zero_index_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_decoded_row_scale. bcf_row_scale_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_table_decoded_row_scale * S ((S (bcf_row_index_lucas_convolution_zero_index_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_index_source) + (bcf_row_scale_lucas_convolution_zero_index_source_table))) /\ ((bcf_row_index_lucas_convolution_zero_index_source_table = 0 /\ (forall bcf_index_lucas_convolution_zero_index_source_table_zero_row. (exists bcf_lt_gap_lucas_convolution_zero_index_source_table_zero_row_bound. bcf_lt_gap_lucas_convolution_zero_index_source_table_zero_row_bound + S (bcf_index_lucas_convolution_zero_index_source_table_zero_row) = S (n)) -> exists bcf_value_lucas_convolution_zero_index_source_table_zero_row. ((((exists bcf_height_lucas_convolution_zero_index_source_table_zero_row_entry. bcf_height_lucas_convolution_zero_index_source_table_zero_row_entry + S (bcf_value_lucas_convolution_zero_index_source_table_zero_row) = S ((S (bcf_index_lucas_convolution_zero_index_source_table_zero_row)) * bcf_row_scale_lucas_convolution_zero_index_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_zero_row_entry. bcf_row_code_lucas_convolution_zero_index_source_table = bcf_quotient_lucas_convolution_zero_index_source_table_zero_row_entry * S ((S (bcf_index_lucas_convolution_zero_index_source_table_zero_row)) * bcf_row_scale_lucas_convolution_zero_index_source_table) + (bcf_value_lucas_convolution_zero_index_source_table_zero_row))) /\ ((bcf_index_lucas_convolution_zero_index_source_table_zero_row = 0 /\ bcf_value_lucas_convolution_zero_index_source_table_zero_row = 1) \/ exists bcf_predecessor_lucas_convolution_zero_index_source_table_zero_row. bcf_index_lucas_convolution_zero_index_source_table_zero_row = S bcf_predecessor_lucas_convolution_zero_index_source_table_zero_row /\ bcf_value_lucas_convolution_zero_index_source_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_convolution_zero_index_source_table bcf_previous_code_lucas_convolution_zero_index_source_table bcf_previous_scale_lucas_convolution_zero_index_source_table. bcf_row_index_lucas_convolution_zero_index_source_table = S bcf_predecessor_lucas_convolution_zero_index_source_table /\ ((((exists bcf_height_lucas_convolution_zero_index_source_table_decoded_previous_code. bcf_height_lucas_convolution_zero_index_source_table_decoded_previous_code + S (bcf_previous_code_lucas_convolution_zero_index_source_table) = S ((S (bcf_predecessor_lucas_convolution_zero_index_source_table)) * bcf_row_code_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_decoded_previous_code. bcf_row_code_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_convolution_zero_index_source_table)) * bcf_row_code_scale_lucas_convolution_zero_index_source) + (bcf_previous_code_lucas_convolution_zero_index_source_table))) /\ ((((exists bcf_height_lucas_convolution_zero_index_source_table_decoded_previous_scale. bcf_height_lucas_convolution_zero_index_source_table_decoded_previous_scale + S (bcf_previous_scale_lucas_convolution_zero_index_source_table) = S ((S (bcf_predecessor_lucas_convolution_zero_index_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_decoded_previous_scale. bcf_row_scale_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_convolution_zero_index_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_index_source) + (bcf_previous_scale_lucas_convolution_zero_index_source_table))) /\ (forall bcf_index_lucas_convolution_zero_index_source_table_row_step. (exists bcf_lt_gap_lucas_convolution_zero_index_source_table_row_step_bound. bcf_lt_gap_lucas_convolution_zero_index_source_table_row_step_bound + S (bcf_index_lucas_convolution_zero_index_source_table_row_step) = S (n)) -> exists bcf_value_lucas_convolution_zero_index_source_table_row_step. ((((exists bcf_height_lucas_convolution_zero_index_source_table_row_step_entry. bcf_height_lucas_convolution_zero_index_source_table_row_step_entry + S (bcf_value_lucas_convolution_zero_index_source_table_row_step) = S ((S (bcf_index_lucas_convolution_zero_index_source_table_row_step)) * bcf_row_scale_lucas_convolution_zero_index_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_row_step_entry. bcf_row_code_lucas_convolution_zero_index_source_table = bcf_quotient_lucas_convolution_zero_index_source_table_row_step_entry * S ((S (bcf_index_lucas_convolution_zero_index_source_table_row_step)) * bcf_row_scale_lucas_convolution_zero_index_source_table) + (bcf_value_lucas_convolution_zero_index_source_table_row_step))) /\ ((bcf_index_lucas_convolution_zero_index_source_table_row_step = 0 /\ bcf_value_lucas_convolution_zero_index_source_table_row_step = 1) \/ exists bcf_predecessor_lucas_convolution_zero_index_source_table_row_step bcf_left_lucas_convolution_zero_index_source_table_row_step bcf_right_lucas_convolution_zero_index_source_table_row_step. bcf_index_lucas_convolution_zero_index_source_table_row_step = S bcf_predecessor_lucas_convolution_zero_index_source_table_row_step /\ ((((exists bcf_height_lucas_convolution_zero_index_source_table_row_step_previous_left. bcf_height_lucas_convolution_zero_index_source_table_row_step_previous_left + S (bcf_left_lucas_convolution_zero_index_source_table_row_step) = S ((S (bcf_predecessor_lucas_convolution_zero_index_source_table_row_step)) * bcf_previous_scale_lucas_convolution_zero_index_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_row_step_previous_left. bcf_previous_code_lucas_convolution_zero_index_source_table = bcf_quotient_lucas_convolution_zero_index_source_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_convolution_zero_index_source_table_row_step)) * bcf_previous_scale_lucas_convolution_zero_index_source_table) + (bcf_left_lucas_convolution_zero_index_source_table_row_step))) /\ ((((exists bcf_height_lucas_convolution_zero_index_source_table_row_step_previous_right. bcf_height_lucas_convolution_zero_index_source_table_row_step_previous_right + S (bcf_right_lucas_convolution_zero_index_source_table_row_step) = S ((S (S (bcf_predecessor_lucas_convolution_zero_index_source_table_row_step))) * bcf_previous_scale_lucas_convolution_zero_index_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_table_row_step_previous_right. bcf_previous_code_lucas_convolution_zero_index_source_table = bcf_quotient_lucas_convolution_zero_index_source_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_convolution_zero_index_source_table_row_step))) * bcf_previous_scale_lucas_convolution_zero_index_source_table) + (bcf_right_lucas_convolution_zero_index_source_table_row_step))) /\ bcf_value_lucas_convolution_zero_index_source_table_row_step = bcf_left_lucas_convolution_zero_index_source_table_row_step + bcf_right_lucas_convolution_zero_index_source_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_convolution_zero_index_source_decoded_row_code. bcf_height_lucas_convolution_zero_index_source_decoded_row_code + S (bcf_row_code_lucas_convolution_zero_index_source) = S ((S (n)) * bcf_row_code_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_decoded_row_code. bcf_row_code_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_decoded_row_code * S ((S (n)) * bcf_row_code_scale_lucas_convolution_zero_index_source) + (bcf_row_code_lucas_convolution_zero_index_source))) /\ ((((exists bcf_height_lucas_convolution_zero_index_source_decoded_row_scale. bcf_height_lucas_convolution_zero_index_source_decoded_row_scale + S (bcf_row_scale_lucas_convolution_zero_index_source) = S ((S (n)) * bcf_row_scale_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_decoded_row_scale. bcf_row_scale_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_lucas_convolution_zero_index_source) + (bcf_row_scale_lucas_convolution_zero_index_source))) /\ (((exists bcf_height_lucas_convolution_zero_index_source_decoded_value. bcf_height_lucas_convolution_zero_index_source_decoded_value + S (C) = S ((S (k)) * bcf_row_scale_lucas_convolution_zero_index_source)) /\ exists bcf_quotient_lucas_convolution_zero_index_source_decoded_value. bcf_row_code_lucas_convolution_zero_index_source = bcf_quotient_lucas_convolution_zero_index_source_decoded_value * S ((S (k)) * bcf_row_scale_lucas_convolution_zero_index_source) + (C))))))))) -> C = 1

Constructive proof overview

Generated structural guide

Every relational binomial coefficient with a witnessed zero lower index is exactly one.

The unchanged tactic script uses 2 declared prerequisites and contains 15 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

LU000O lucas_choose_lower_eq_transport choose_zero Alpha theorem; checked-use authorized

Direct dependents

LU000W lucas_prime_shift_below_base LU000Z lucas_prime_shift_high_column

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

15 script commands · 2 reading checkpoints · 0 local claims

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.

Named ingredients (1)
lucas choose lower eq transport
01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
  2. L2
    intro k
  3. L3
    intro C
  4. L4
    intro hzero
  5. L5
    intro hchoose
02Use earlier factsL6–15

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L6
    specialize choose_zero n
  2. L7
    specialize choose_zero C
  3. L8
    apply choose_zero
  4. L9
    specialize lucas_choose_lower_eq_transport n
  5. L10
    specialize lucas_choose_lower_eq_transport k
  6. L11
    specialize lucas_choose_lower_eq_transport 0
  7. L12
    specialize lucas_choose_lower_eq_transport C
  8. L13
    apply lucas_choose_lower_eq_transport
  9. L14
    exact hzero
  10. L15
    exact hchoose

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro n
  2. 0002intro k
  3. 0003intro C
  4. 0004intro hzero
  5. 0005intro hchoose
  6. 0006specialize choose_zero n
  7. 0007specialize choose_zero C
  8. 0008apply choose_zero
  9. 0009specialize lucas_choose_lower_eq_transport n
  10. 0010specialize lucas_choose_lower_eq_transport k
  11. 0011specialize lucas_choose_lower_eq_transport 0
  12. 0012specialize lucas_choose_lower_eq_transport C
  13. 0013apply lucas_choose_lower_eq_transport
  14. 0014exact hzero
  15. 0015exact hchoose