LU001E · theorem body

lucas_choose_prefix_empty

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

Any pair of beta-coded source streams has a constructively empty relational-Choose coefficient prefix.

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

∀ qb. ∀ qc. ∀ db. ∀ dc. ∃ z. ∃ t. ∀ x. Lt(x,0) → ∃ y. ∃ n. ∃ m. BetaAt(qb,qc,x,y) ∧ (BetaAt(db,dc,x,n) ∧ (BetaAt(z,t,x,m) ∧ Choose(y,n,m)))

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

PD0002 Lt PD0013 BetaAt PD0041 Choose

In local proof propositions

none
Exact expanded first-order statement
forall qb qc db dc. exists z t. (forall lmd_choose_index_choose_empty. (exists lmd_gap_choose_empty_bound. lmd_gap_choose_empty_bound + S (lmd_choose_index_choose_empty) = (0)) -> exists lmd_choose_upper_choose_empty lmd_choose_lower_choose_empty lmd_choose_value_choose_empty. ((((exists ff_h_lmd_choose_empty_upper. ff_h_lmd_choose_empty_upper + S (lmd_choose_upper_choose_empty) = S ((S (lmd_choose_index_choose_empty)) * qc)) /\ exists ff_q_lmd_choose_empty_upper. qb = ff_q_lmd_choose_empty_upper * S ((S (lmd_choose_index_choose_empty)) * qc) + (lmd_choose_upper_choose_empty))) /\ ((((exists ff_h_lmd_choose_empty_lower. ff_h_lmd_choose_empty_lower + S (lmd_choose_lower_choose_empty) = S ((S (lmd_choose_index_choose_empty)) * dc)) /\ exists ff_q_lmd_choose_empty_lower. db = ff_q_lmd_choose_empty_lower * S ((S (lmd_choose_index_choose_empty)) * dc) + (lmd_choose_lower_choose_empty))) /\ ((((exists ff_h_lmd_choose_empty_result. ff_h_lmd_choose_empty_result + S (lmd_choose_value_choose_empty) = S ((S (lmd_choose_index_choose_empty)) * t)) /\ exists ff_q_lmd_choose_empty_result. z = ff_q_lmd_choose_empty_result * S ((S (lmd_choose_index_choose_empty)) * t) + (lmd_choose_value_choose_empty))) /\ (((exists bcf_lt_gap_lmd_choose_empty_choose_out_of_range. bcf_lt_gap_lmd_choose_empty_choose_out_of_range + S (lmd_choose_upper_choose_empty) = lmd_choose_lower_choose_empty) /\ lmd_choose_value_choose_empty = 0) \/ ((exists bcf_le_gap_lmd_choose_empty_choose_in_range. bcf_le_gap_lmd_choose_empty_choose_in_range + (lmd_choose_lower_choose_empty) = lmd_choose_upper_choose_empty) /\ (exists bcf_row_code_code_lmd_choose_empty_choose bcf_row_code_scale_lmd_choose_empty_choose bcf_row_scale_code_lmd_choose_empty_choose bcf_row_scale_scale_lmd_choose_empty_choose bcf_row_code_lmd_choose_empty_choose bcf_row_scale_lmd_choose_empty_choose. ((forall bcf_row_index_lmd_choose_empty_choose_table. (exists bcf_lt_gap_lmd_choose_empty_choose_table_row_bound. bcf_lt_gap_lmd_choose_empty_choose_table_row_bound + S (bcf_row_index_lmd_choose_empty_choose_table) = S (lmd_choose_upper_choose_empty)) -> exists bcf_row_code_lmd_choose_empty_choose_table bcf_row_scale_lmd_choose_empty_choose_table. ((((exists bcf_height_lmd_choose_empty_choose_table_decoded_row_code. bcf_height_lmd_choose_empty_choose_table_decoded_row_code + S (bcf_row_code_lmd_choose_empty_choose_table) = S ((S (bcf_row_index_lmd_choose_empty_choose_table)) * bcf_row_code_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_decoded_row_code. bcf_row_code_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_table_decoded_row_code * S ((S (bcf_row_index_lmd_choose_empty_choose_table)) * bcf_row_code_scale_lmd_choose_empty_choose) + (bcf_row_code_lmd_choose_empty_choose_table))) /\ ((((exists bcf_height_lmd_choose_empty_choose_table_decoded_row_scale. bcf_height_lmd_choose_empty_choose_table_decoded_row_scale + S (bcf_row_scale_lmd_choose_empty_choose_table) = S ((S (bcf_row_index_lmd_choose_empty_choose_table)) * bcf_row_scale_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_decoded_row_scale. bcf_row_scale_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_table_decoded_row_scale * S ((S (bcf_row_index_lmd_choose_empty_choose_table)) * bcf_row_scale_scale_lmd_choose_empty_choose) + (bcf_row_scale_lmd_choose_empty_choose_table))) /\ ((bcf_row_index_lmd_choose_empty_choose_table = 0 /\ (forall bcf_index_lmd_choose_empty_choose_table_zero_row. (exists bcf_lt_gap_lmd_choose_empty_choose_table_zero_row_bound. bcf_lt_gap_lmd_choose_empty_choose_table_zero_row_bound + S (bcf_index_lmd_choose_empty_choose_table_zero_row) = S (lmd_choose_upper_choose_empty)) -> exists bcf_value_lmd_choose_empty_choose_table_zero_row. ((((exists bcf_height_lmd_choose_empty_choose_table_zero_row_entry. bcf_height_lmd_choose_empty_choose_table_zero_row_entry + S (bcf_value_lmd_choose_empty_choose_table_zero_row) = S ((S (bcf_index_lmd_choose_empty_choose_table_zero_row)) * bcf_row_scale_lmd_choose_empty_choose_table)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_zero_row_entry. bcf_row_code_lmd_choose_empty_choose_table = bcf_quotient_lmd_choose_empty_choose_table_zero_row_entry * S ((S (bcf_index_lmd_choose_empty_choose_table_zero_row)) * bcf_row_scale_lmd_choose_empty_choose_table) + (bcf_value_lmd_choose_empty_choose_table_zero_row))) /\ ((bcf_index_lmd_choose_empty_choose_table_zero_row = 0 /\ bcf_value_lmd_choose_empty_choose_table_zero_row = 1) \/ exists bcf_predecessor_lmd_choose_empty_choose_table_zero_row. bcf_index_lmd_choose_empty_choose_table_zero_row = S bcf_predecessor_lmd_choose_empty_choose_table_zero_row /\ bcf_value_lmd_choose_empty_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_lmd_choose_empty_choose_table bcf_previous_code_lmd_choose_empty_choose_table bcf_previous_scale_lmd_choose_empty_choose_table. bcf_row_index_lmd_choose_empty_choose_table = S bcf_predecessor_lmd_choose_empty_choose_table /\ ((((exists bcf_height_lmd_choose_empty_choose_table_decoded_previous_code. bcf_height_lmd_choose_empty_choose_table_decoded_previous_code + S (bcf_previous_code_lmd_choose_empty_choose_table) = S ((S (bcf_predecessor_lmd_choose_empty_choose_table)) * bcf_row_code_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_decoded_previous_code. bcf_row_code_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_table_decoded_previous_code * S ((S (bcf_predecessor_lmd_choose_empty_choose_table)) * bcf_row_code_scale_lmd_choose_empty_choose) + (bcf_previous_code_lmd_choose_empty_choose_table))) /\ ((((exists bcf_height_lmd_choose_empty_choose_table_decoded_previous_scale. bcf_height_lmd_choose_empty_choose_table_decoded_previous_scale + S (bcf_previous_scale_lmd_choose_empty_choose_table) = S ((S (bcf_predecessor_lmd_choose_empty_choose_table)) * bcf_row_scale_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_decoded_previous_scale. bcf_row_scale_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_lmd_choose_empty_choose_table)) * bcf_row_scale_scale_lmd_choose_empty_choose) + (bcf_previous_scale_lmd_choose_empty_choose_table))) /\ (forall bcf_index_lmd_choose_empty_choose_table_row_step. (exists bcf_lt_gap_lmd_choose_empty_choose_table_row_step_bound. bcf_lt_gap_lmd_choose_empty_choose_table_row_step_bound + S (bcf_index_lmd_choose_empty_choose_table_row_step) = S (lmd_choose_upper_choose_empty)) -> exists bcf_value_lmd_choose_empty_choose_table_row_step. ((((exists bcf_height_lmd_choose_empty_choose_table_row_step_entry. bcf_height_lmd_choose_empty_choose_table_row_step_entry + S (bcf_value_lmd_choose_empty_choose_table_row_step) = S ((S (bcf_index_lmd_choose_empty_choose_table_row_step)) * bcf_row_scale_lmd_choose_empty_choose_table)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_row_step_entry. bcf_row_code_lmd_choose_empty_choose_table = bcf_quotient_lmd_choose_empty_choose_table_row_step_entry * S ((S (bcf_index_lmd_choose_empty_choose_table_row_step)) * bcf_row_scale_lmd_choose_empty_choose_table) + (bcf_value_lmd_choose_empty_choose_table_row_step))) /\ ((bcf_index_lmd_choose_empty_choose_table_row_step = 0 /\ bcf_value_lmd_choose_empty_choose_table_row_step = 1) \/ exists bcf_predecessor_lmd_choose_empty_choose_table_row_step bcf_left_lmd_choose_empty_choose_table_row_step bcf_right_lmd_choose_empty_choose_table_row_step. bcf_index_lmd_choose_empty_choose_table_row_step = S bcf_predecessor_lmd_choose_empty_choose_table_row_step /\ ((((exists bcf_height_lmd_choose_empty_choose_table_row_step_previous_left. bcf_height_lmd_choose_empty_choose_table_row_step_previous_left + S (bcf_left_lmd_choose_empty_choose_table_row_step) = S ((S (bcf_predecessor_lmd_choose_empty_choose_table_row_step)) * bcf_previous_scale_lmd_choose_empty_choose_table)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_row_step_previous_left. bcf_previous_code_lmd_choose_empty_choose_table = bcf_quotient_lmd_choose_empty_choose_table_row_step_previous_left * S ((S (bcf_predecessor_lmd_choose_empty_choose_table_row_step)) * bcf_previous_scale_lmd_choose_empty_choose_table) + (bcf_left_lmd_choose_empty_choose_table_row_step))) /\ ((((exists bcf_height_lmd_choose_empty_choose_table_row_step_previous_right. bcf_height_lmd_choose_empty_choose_table_row_step_previous_right + S (bcf_right_lmd_choose_empty_choose_table_row_step) = S ((S (S (bcf_predecessor_lmd_choose_empty_choose_table_row_step))) * bcf_previous_scale_lmd_choose_empty_choose_table)) /\ exists bcf_quotient_lmd_choose_empty_choose_table_row_step_previous_right. bcf_previous_code_lmd_choose_empty_choose_table = bcf_quotient_lmd_choose_empty_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_lmd_choose_empty_choose_table_row_step))) * bcf_previous_scale_lmd_choose_empty_choose_table) + (bcf_right_lmd_choose_empty_choose_table_row_step))) /\ bcf_value_lmd_choose_empty_choose_table_row_step = bcf_left_lmd_choose_empty_choose_table_row_step + bcf_right_lmd_choose_empty_choose_table_row_step))))))))))) /\ ((((exists bcf_height_lmd_choose_empty_choose_decoded_row_code. bcf_height_lmd_choose_empty_choose_decoded_row_code + S (bcf_row_code_lmd_choose_empty_choose) = S ((S (lmd_choose_upper_choose_empty)) * bcf_row_code_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_decoded_row_code. bcf_row_code_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_decoded_row_code * S ((S (lmd_choose_upper_choose_empty)) * bcf_row_code_scale_lmd_choose_empty_choose) + (bcf_row_code_lmd_choose_empty_choose))) /\ ((((exists bcf_height_lmd_choose_empty_choose_decoded_row_scale. bcf_height_lmd_choose_empty_choose_decoded_row_scale + S (bcf_row_scale_lmd_choose_empty_choose) = S ((S (lmd_choose_upper_choose_empty)) * bcf_row_scale_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_decoded_row_scale. bcf_row_scale_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_decoded_row_scale * S ((S (lmd_choose_upper_choose_empty)) * bcf_row_scale_scale_lmd_choose_empty_choose) + (bcf_row_scale_lmd_choose_empty_choose))) /\ (((exists bcf_height_lmd_choose_empty_choose_decoded_value. bcf_height_lmd_choose_empty_choose_decoded_value + S (lmd_choose_value_choose_empty) = S ((S (lmd_choose_lower_choose_empty)) * bcf_row_scale_lmd_choose_empty_choose)) /\ exists bcf_quotient_lmd_choose_empty_choose_decoded_value. bcf_row_code_lmd_choose_empty_choose = bcf_quotient_lmd_choose_empty_choose_decoded_value * S ((S (lmd_choose_lower_choose_empty)) * bcf_row_scale_lmd_choose_empty_choose) + (lmd_choose_value_choose_empty)))))))))))))

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

LU001G lucas_choose_prefix_exists

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

13 script commands · 6 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro qb
  2. L2
    intro qc
  3. L3
    intro db
  4. L4
    intro dc
02Construct an explicit witnessL5–6

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

  1. L5
    exists 0
  2. L6
    exists 0
03Fix variables and assumptionsL7–8

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

  1. L7
    intro i
  2. L8
    intro hi
04Separate the logical casesL9–10

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

  1. L9
    exfalso
  2. L10
    cases hi
05Calculate and transport equalitiesL11–11

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

  1. L11
    rewrite PA4 at hi_witness
06Use earlier factsL12–13

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

  1. L12
    apply PA1
  2. L13
    exact hi_witness

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro qb
  2. 0002intro qc
  3. 0003intro db
  4. 0004intro dc
  5. 0005exists 0
  6. 0006exists 0
  7. 0007intro i
  8. 0008intro hi
  9. 0009exfalso
  10. 0010cases hi
  11. 0011rewrite PA4 at hi_witness
  12. 0012apply PA1
  13. 0013exact hi_witness