LU000P · theorem body

lucas_choose_zero_index_is_one

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

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

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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

∀ n. ∀ k. ∀ C. k = 0 → Choose(n,k,C) → C = 1

Every 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

PD0041 Choose

In local proof propositions

none
Exact expanded first-order 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

Proof neighborhood

Direct theorem prerequisites

LU000O lucas_choose_lower_eq_transport choose_zero · Alpha closed

Direct theorem dependents

LU000W lucas_prime_shift_below_base LU000Z lucas_prime_shift_high_column

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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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