LU000Q · theorem body

lucas_choose_zero_upper_positive_is_zero

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

A positive lower index forces the zeroth Pascal-row coefficient to vanish.

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

∀ k. ∀ C. ¬k = 0 → Choose(0,k,C) → C = 0

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

PD0002 Lt
Exact expanded first-order statement
forall k C. ~(k = 0) -> (((exists bcf_lt_gap_lucas_convolution_zero_upper_source_out_of_range. bcf_lt_gap_lucas_convolution_zero_upper_source_out_of_range + S (0) = k) /\ C = 0) \/ ((exists bcf_le_gap_lucas_convolution_zero_upper_source_in_range. bcf_le_gap_lucas_convolution_zero_upper_source_in_range + (k) = 0) /\ (exists bcf_row_code_code_lucas_convolution_zero_upper_source bcf_row_code_scale_lucas_convolution_zero_upper_source bcf_row_scale_code_lucas_convolution_zero_upper_source bcf_row_scale_scale_lucas_convolution_zero_upper_source bcf_row_code_lucas_convolution_zero_upper_source bcf_row_scale_lucas_convolution_zero_upper_source. ((forall bcf_row_index_lucas_convolution_zero_upper_source_table. (exists bcf_lt_gap_lucas_convolution_zero_upper_source_table_row_bound. bcf_lt_gap_lucas_convolution_zero_upper_source_table_row_bound + S (bcf_row_index_lucas_convolution_zero_upper_source_table) = S (0)) -> exists bcf_row_code_lucas_convolution_zero_upper_source_table bcf_row_scale_lucas_convolution_zero_upper_source_table. ((((exists bcf_height_lucas_convolution_zero_upper_source_table_decoded_row_code. bcf_height_lucas_convolution_zero_upper_source_table_decoded_row_code + S (bcf_row_code_lucas_convolution_zero_upper_source_table) = S ((S (bcf_row_index_lucas_convolution_zero_upper_source_table)) * bcf_row_code_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_row_code. bcf_row_code_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_row_code * S ((S (bcf_row_index_lucas_convolution_zero_upper_source_table)) * bcf_row_code_scale_lucas_convolution_zero_upper_source) + (bcf_row_code_lucas_convolution_zero_upper_source_table))) /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_table_decoded_row_scale. bcf_height_lucas_convolution_zero_upper_source_table_decoded_row_scale + S (bcf_row_scale_lucas_convolution_zero_upper_source_table) = S ((S (bcf_row_index_lucas_convolution_zero_upper_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_row_scale. bcf_row_scale_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_row_scale * S ((S (bcf_row_index_lucas_convolution_zero_upper_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_upper_source) + (bcf_row_scale_lucas_convolution_zero_upper_source_table))) /\ ((bcf_row_index_lucas_convolution_zero_upper_source_table = 0 /\ (forall bcf_index_lucas_convolution_zero_upper_source_table_zero_row. (exists bcf_lt_gap_lucas_convolution_zero_upper_source_table_zero_row_bound. bcf_lt_gap_lucas_convolution_zero_upper_source_table_zero_row_bound + S (bcf_index_lucas_convolution_zero_upper_source_table_zero_row) = S (0)) -> exists bcf_value_lucas_convolution_zero_upper_source_table_zero_row. ((((exists bcf_height_lucas_convolution_zero_upper_source_table_zero_row_entry. bcf_height_lucas_convolution_zero_upper_source_table_zero_row_entry + S (bcf_value_lucas_convolution_zero_upper_source_table_zero_row) = S ((S (bcf_index_lucas_convolution_zero_upper_source_table_zero_row)) * bcf_row_scale_lucas_convolution_zero_upper_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_zero_row_entry. bcf_row_code_lucas_convolution_zero_upper_source_table = bcf_quotient_lucas_convolution_zero_upper_source_table_zero_row_entry * S ((S (bcf_index_lucas_convolution_zero_upper_source_table_zero_row)) * bcf_row_scale_lucas_convolution_zero_upper_source_table) + (bcf_value_lucas_convolution_zero_upper_source_table_zero_row))) /\ ((bcf_index_lucas_convolution_zero_upper_source_table_zero_row = 0 /\ bcf_value_lucas_convolution_zero_upper_source_table_zero_row = 1) \/ exists bcf_predecessor_lucas_convolution_zero_upper_source_table_zero_row. bcf_index_lucas_convolution_zero_upper_source_table_zero_row = S bcf_predecessor_lucas_convolution_zero_upper_source_table_zero_row /\ bcf_value_lucas_convolution_zero_upper_source_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_convolution_zero_upper_source_table bcf_previous_code_lucas_convolution_zero_upper_source_table bcf_previous_scale_lucas_convolution_zero_upper_source_table. bcf_row_index_lucas_convolution_zero_upper_source_table = S bcf_predecessor_lucas_convolution_zero_upper_source_table /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_table_decoded_previous_code. bcf_height_lucas_convolution_zero_upper_source_table_decoded_previous_code + S (bcf_previous_code_lucas_convolution_zero_upper_source_table) = S ((S (bcf_predecessor_lucas_convolution_zero_upper_source_table)) * bcf_row_code_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_previous_code. bcf_row_code_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_convolution_zero_upper_source_table)) * bcf_row_code_scale_lucas_convolution_zero_upper_source) + (bcf_previous_code_lucas_convolution_zero_upper_source_table))) /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_table_decoded_previous_scale. bcf_height_lucas_convolution_zero_upper_source_table_decoded_previous_scale + S (bcf_previous_scale_lucas_convolution_zero_upper_source_table) = S ((S (bcf_predecessor_lucas_convolution_zero_upper_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_previous_scale. bcf_row_scale_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_convolution_zero_upper_source_table)) * bcf_row_scale_scale_lucas_convolution_zero_upper_source) + (bcf_previous_scale_lucas_convolution_zero_upper_source_table))) /\ (forall bcf_index_lucas_convolution_zero_upper_source_table_row_step. (exists bcf_lt_gap_lucas_convolution_zero_upper_source_table_row_step_bound. bcf_lt_gap_lucas_convolution_zero_upper_source_table_row_step_bound + S (bcf_index_lucas_convolution_zero_upper_source_table_row_step) = S (0)) -> exists bcf_value_lucas_convolution_zero_upper_source_table_row_step. ((((exists bcf_height_lucas_convolution_zero_upper_source_table_row_step_entry. bcf_height_lucas_convolution_zero_upper_source_table_row_step_entry + S (bcf_value_lucas_convolution_zero_upper_source_table_row_step) = S ((S (bcf_index_lucas_convolution_zero_upper_source_table_row_step)) * bcf_row_scale_lucas_convolution_zero_upper_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_row_step_entry. bcf_row_code_lucas_convolution_zero_upper_source_table = bcf_quotient_lucas_convolution_zero_upper_source_table_row_step_entry * S ((S (bcf_index_lucas_convolution_zero_upper_source_table_row_step)) * bcf_row_scale_lucas_convolution_zero_upper_source_table) + (bcf_value_lucas_convolution_zero_upper_source_table_row_step))) /\ ((bcf_index_lucas_convolution_zero_upper_source_table_row_step = 0 /\ bcf_value_lucas_convolution_zero_upper_source_table_row_step = 1) \/ exists bcf_predecessor_lucas_convolution_zero_upper_source_table_row_step bcf_left_lucas_convolution_zero_upper_source_table_row_step bcf_right_lucas_convolution_zero_upper_source_table_row_step. bcf_index_lucas_convolution_zero_upper_source_table_row_step = S bcf_predecessor_lucas_convolution_zero_upper_source_table_row_step /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_table_row_step_previous_left. bcf_height_lucas_convolution_zero_upper_source_table_row_step_previous_left + S (bcf_left_lucas_convolution_zero_upper_source_table_row_step) = S ((S (bcf_predecessor_lucas_convolution_zero_upper_source_table_row_step)) * bcf_previous_scale_lucas_convolution_zero_upper_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_row_step_previous_left. bcf_previous_code_lucas_convolution_zero_upper_source_table = bcf_quotient_lucas_convolution_zero_upper_source_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_convolution_zero_upper_source_table_row_step)) * bcf_previous_scale_lucas_convolution_zero_upper_source_table) + (bcf_left_lucas_convolution_zero_upper_source_table_row_step))) /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_table_row_step_previous_right. bcf_height_lucas_convolution_zero_upper_source_table_row_step_previous_right + S (bcf_right_lucas_convolution_zero_upper_source_table_row_step) = S ((S (S (bcf_predecessor_lucas_convolution_zero_upper_source_table_row_step))) * bcf_previous_scale_lucas_convolution_zero_upper_source_table)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_table_row_step_previous_right. bcf_previous_code_lucas_convolution_zero_upper_source_table = bcf_quotient_lucas_convolution_zero_upper_source_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_convolution_zero_upper_source_table_row_step))) * bcf_previous_scale_lucas_convolution_zero_upper_source_table) + (bcf_right_lucas_convolution_zero_upper_source_table_row_step))) /\ bcf_value_lucas_convolution_zero_upper_source_table_row_step = bcf_left_lucas_convolution_zero_upper_source_table_row_step + bcf_right_lucas_convolution_zero_upper_source_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_decoded_row_code. bcf_height_lucas_convolution_zero_upper_source_decoded_row_code + S (bcf_row_code_lucas_convolution_zero_upper_source) = S ((S (0)) * bcf_row_code_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_decoded_row_code. bcf_row_code_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_decoded_row_code * S ((S (0)) * bcf_row_code_scale_lucas_convolution_zero_upper_source) + (bcf_row_code_lucas_convolution_zero_upper_source))) /\ ((((exists bcf_height_lucas_convolution_zero_upper_source_decoded_row_scale. bcf_height_lucas_convolution_zero_upper_source_decoded_row_scale + S (bcf_row_scale_lucas_convolution_zero_upper_source) = S ((S (0)) * bcf_row_scale_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_decoded_row_scale. bcf_row_scale_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_decoded_row_scale * S ((S (0)) * bcf_row_scale_scale_lucas_convolution_zero_upper_source) + (bcf_row_scale_lucas_convolution_zero_upper_source))) /\ (((exists bcf_height_lucas_convolution_zero_upper_source_decoded_value. bcf_height_lucas_convolution_zero_upper_source_decoded_value + S (C) = S ((S (k)) * bcf_row_scale_lucas_convolution_zero_upper_source)) /\ exists bcf_quotient_lucas_convolution_zero_upper_source_decoded_value. bcf_row_code_lucas_convolution_zero_upper_source = bcf_quotient_lucas_convolution_zero_upper_source_decoded_value * S ((S (k)) * bcf_row_scale_lucas_convolution_zero_upper_source) + (C))))))))) -> C = 0

Proof neighborhood

Direct theorem prerequisites

nonzero_is_succ · Stable closed choose_out_of_range_zero · Alpha closed

Direct theorem dependents

LU000M lucas_prime_block_digit_congruence 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

19 script commands · 7 reading checkpoints · 2 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro k
  2. L2
    intro C
  3. L3
    intro hnonzero
  4. L4
    intro hchoose
02Establish hsuccessorL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L5
    have hsuccessor : exists a. k = S a
  2. L6
    specialize nonzero_is_succ k
  3. L7
    apply nonzero_is_succ
  4. L8
    exact hnonzero
03Separate the logical casesL9–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases hsuccessor
04Establish hpositiveL10–10

Establish this local claim before using it. It is not an additional assumption.

  1. L10
    have hpositive : Lt(0,k)Definitions: Lt(0,k)Original native command in the exact edition
05Construct an explicit witnessL11–11

Supply the displayed value, then prove that it has the required property.

  1. L11
    exists x
06Calculate and transport equalitiesL12–13

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L12
    rewrite hsuccessor_witness
  2. L13
    simp
07Use earlier factsL14–19

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

  1. L14
    specialize choose_out_of_range_zero 0
  2. L15
    specialize choose_out_of_range_zero k
  3. L16
    specialize choose_out_of_range_zero C
  4. L17
    apply choose_out_of_range_zero
  5. L18
    exact hpositive
  6. L19
    exact hchoose

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro k
  2. 0002intro C
  3. 0003intro hnonzero
  4. 0004intro hchoose
  5. 0005have hsuccessor : exists a. k = S a
  6. 0006specialize nonzero_is_succ k
  7. 0007apply nonzero_is_succ
  8. 0008exact hnonzero
  9. 0009cases hsuccessor
  10. 0010have hpositive : Lt(0,k)
    Exact native replay linehave hpositive : exists lcv_gap_zero_upper_positive. lcv_gap_zero_upper_positive + S (0) = (k)
  11. 0011exists x
  12. 0012rewrite hsuccessor_witness
  13. 0013simp
  14. 0014specialize choose_out_of_range_zero 0
  15. 0015specialize choose_out_of_range_zero k
  16. 0016specialize choose_out_of_range_zero C
  17. 0017apply choose_out_of_range_zero
  18. 0018exact hpositive
  19. 0019exact hchoose