LU000L

lucas_zero_upper_quotient_high_column_vanishes

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

The zero-upper-quotient boundary of the Lucas prime block is constructively zero in every positive lower-quotient column.

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 p r d e C. (exists lbd_gap_zero_upper_bound. lbd_gap_zero_upper_bound + S (d) = (p)) -> ~(r = 0) -> (((exists bcf_lt_gap_lucas_block_digit_zero_upper_choose_out_of_range. bcf_lt_gap_lucas_block_digit_zero_upper_choose_out_of_range + S (p * 0 + d) = p * r + e) /\ C = 0) \/ ((exists bcf_le_gap_lucas_block_digit_zero_upper_choose_in_range. bcf_le_gap_lucas_block_digit_zero_upper_choose_in_range + (p * r + e) = p * 0 + d) /\ (exists bcf_row_code_code_lucas_block_digit_zero_upper_choose bcf_row_code_scale_lucas_block_digit_zero_upper_choose bcf_row_scale_code_lucas_block_digit_zero_upper_choose bcf_row_scale_scale_lucas_block_digit_zero_upper_choose bcf_row_code_lucas_block_digit_zero_upper_choose bcf_row_scale_lucas_block_digit_zero_upper_choose. ((forall bcf_row_index_lucas_block_digit_zero_upper_choose_table. (exists bcf_lt_gap_lucas_block_digit_zero_upper_choose_table_row_bound. bcf_lt_gap_lucas_block_digit_zero_upper_choose_table_row_bound + S (bcf_row_index_lucas_block_digit_zero_upper_choose_table) = S (p * 0 + d)) -> exists bcf_row_code_lucas_block_digit_zero_upper_choose_table bcf_row_scale_lucas_block_digit_zero_upper_choose_table. ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_row_code. bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_row_code + S (bcf_row_code_lucas_block_digit_zero_upper_choose_table) = S ((S (bcf_row_index_lucas_block_digit_zero_upper_choose_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_row_code. bcf_row_code_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_row_code * S ((S (bcf_row_index_lucas_block_digit_zero_upper_choose_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_choose) + (bcf_row_code_lucas_block_digit_zero_upper_choose_table))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_row_scale. bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_zero_upper_choose_table) = S ((S (bcf_row_index_lucas_block_digit_zero_upper_choose_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_row_scale * S ((S (bcf_row_index_lucas_block_digit_zero_upper_choose_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_choose) + (bcf_row_scale_lucas_block_digit_zero_upper_choose_table))) /\ ((bcf_row_index_lucas_block_digit_zero_upper_choose_table = 0 /\ (forall bcf_index_lucas_block_digit_zero_upper_choose_table_zero_row. (exists bcf_lt_gap_lucas_block_digit_zero_upper_choose_table_zero_row_bound. bcf_lt_gap_lucas_block_digit_zero_upper_choose_table_zero_row_bound + S (bcf_index_lucas_block_digit_zero_upper_choose_table_zero_row) = S (p * 0 + d)) -> exists bcf_value_lucas_block_digit_zero_upper_choose_table_zero_row. ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_zero_row_entry. bcf_height_lucas_block_digit_zero_upper_choose_table_zero_row_entry + S (bcf_value_lucas_block_digit_zero_upper_choose_table_zero_row) = S ((S (bcf_index_lucas_block_digit_zero_upper_choose_table_zero_row)) * bcf_row_scale_lucas_block_digit_zero_upper_choose_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_zero_row_entry. bcf_row_code_lucas_block_digit_zero_upper_choose_table = bcf_quotient_lucas_block_digit_zero_upper_choose_table_zero_row_entry * S ((S (bcf_index_lucas_block_digit_zero_upper_choose_table_zero_row)) * bcf_row_scale_lucas_block_digit_zero_upper_choose_table) + (bcf_value_lucas_block_digit_zero_upper_choose_table_zero_row))) /\ ((bcf_index_lucas_block_digit_zero_upper_choose_table_zero_row = 0 /\ bcf_value_lucas_block_digit_zero_upper_choose_table_zero_row = 1) \/ exists bcf_predecessor_lucas_block_digit_zero_upper_choose_table_zero_row. bcf_index_lucas_block_digit_zero_upper_choose_table_zero_row = S bcf_predecessor_lucas_block_digit_zero_upper_choose_table_zero_row /\ bcf_value_lucas_block_digit_zero_upper_choose_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_block_digit_zero_upper_choose_table bcf_previous_code_lucas_block_digit_zero_upper_choose_table bcf_previous_scale_lucas_block_digit_zero_upper_choose_table. bcf_row_index_lucas_block_digit_zero_upper_choose_table = S bcf_predecessor_lucas_block_digit_zero_upper_choose_table /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_previous_code. bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_previous_code + S (bcf_previous_code_lucas_block_digit_zero_upper_choose_table) = S ((S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_previous_code. bcf_row_code_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_choose) + (bcf_previous_code_lucas_block_digit_zero_upper_choose_table))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_previous_scale. bcf_height_lucas_block_digit_zero_upper_choose_table_decoded_previous_scale + S (bcf_previous_scale_lucas_block_digit_zero_upper_choose_table) = S ((S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_previous_scale. bcf_row_scale_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_choose) + (bcf_previous_scale_lucas_block_digit_zero_upper_choose_table))) /\ (forall bcf_index_lucas_block_digit_zero_upper_choose_table_row_step. (exists bcf_lt_gap_lucas_block_digit_zero_upper_choose_table_row_step_bound. bcf_lt_gap_lucas_block_digit_zero_upper_choose_table_row_step_bound + S (bcf_index_lucas_block_digit_zero_upper_choose_table_row_step) = S (p * 0 + d)) -> exists bcf_value_lucas_block_digit_zero_upper_choose_table_row_step. ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_row_step_entry. bcf_height_lucas_block_digit_zero_upper_choose_table_row_step_entry + S (bcf_value_lucas_block_digit_zero_upper_choose_table_row_step) = S ((S (bcf_index_lucas_block_digit_zero_upper_choose_table_row_step)) * bcf_row_scale_lucas_block_digit_zero_upper_choose_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_row_step_entry. bcf_row_code_lucas_block_digit_zero_upper_choose_table = bcf_quotient_lucas_block_digit_zero_upper_choose_table_row_step_entry * S ((S (bcf_index_lucas_block_digit_zero_upper_choose_table_row_step)) * bcf_row_scale_lucas_block_digit_zero_upper_choose_table) + (bcf_value_lucas_block_digit_zero_upper_choose_table_row_step))) /\ ((bcf_index_lucas_block_digit_zero_upper_choose_table_row_step = 0 /\ bcf_value_lucas_block_digit_zero_upper_choose_table_row_step = 1) \/ exists bcf_predecessor_lucas_block_digit_zero_upper_choose_table_row_step bcf_left_lucas_block_digit_zero_upper_choose_table_row_step bcf_right_lucas_block_digit_zero_upper_choose_table_row_step. bcf_index_lucas_block_digit_zero_upper_choose_table_row_step = S bcf_predecessor_lucas_block_digit_zero_upper_choose_table_row_step /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_row_step_previous_left. bcf_height_lucas_block_digit_zero_upper_choose_table_row_step_previous_left + S (bcf_left_lucas_block_digit_zero_upper_choose_table_row_step) = S ((S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table_row_step)) * bcf_previous_scale_lucas_block_digit_zero_upper_choose_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_row_step_previous_left. bcf_previous_code_lucas_block_digit_zero_upper_choose_table = bcf_quotient_lucas_block_digit_zero_upper_choose_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table_row_step)) * bcf_previous_scale_lucas_block_digit_zero_upper_choose_table) + (bcf_left_lucas_block_digit_zero_upper_choose_table_row_step))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_table_row_step_previous_right. bcf_height_lucas_block_digit_zero_upper_choose_table_row_step_previous_right + S (bcf_right_lucas_block_digit_zero_upper_choose_table_row_step) = S ((S (S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table_row_step))) * bcf_previous_scale_lucas_block_digit_zero_upper_choose_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_table_row_step_previous_right. bcf_previous_code_lucas_block_digit_zero_upper_choose_table = bcf_quotient_lucas_block_digit_zero_upper_choose_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_block_digit_zero_upper_choose_table_row_step))) * bcf_previous_scale_lucas_block_digit_zero_upper_choose_table) + (bcf_right_lucas_block_digit_zero_upper_choose_table_row_step))) /\ bcf_value_lucas_block_digit_zero_upper_choose_table_row_step = bcf_left_lucas_block_digit_zero_upper_choose_table_row_step + bcf_right_lucas_block_digit_zero_upper_choose_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_decoded_row_code. bcf_height_lucas_block_digit_zero_upper_choose_decoded_row_code + S (bcf_row_code_lucas_block_digit_zero_upper_choose) = S ((S (p * 0 + d)) * bcf_row_code_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_decoded_row_code. bcf_row_code_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_decoded_row_code * S ((S (p * 0 + d)) * bcf_row_code_scale_lucas_block_digit_zero_upper_choose) + (bcf_row_code_lucas_block_digit_zero_upper_choose))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_choose_decoded_row_scale. bcf_height_lucas_block_digit_zero_upper_choose_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_zero_upper_choose) = S ((S (p * 0 + d)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_decoded_row_scale * S ((S (p * 0 + d)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_choose) + (bcf_row_scale_lucas_block_digit_zero_upper_choose))) /\ (((exists bcf_height_lucas_block_digit_zero_upper_choose_decoded_value. bcf_height_lucas_block_digit_zero_upper_choose_decoded_value + S (C) = S ((S (p * r + e)) * bcf_row_scale_lucas_block_digit_zero_upper_choose)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_choose_decoded_value. bcf_row_code_lucas_block_digit_zero_upper_choose = bcf_quotient_lucas_block_digit_zero_upper_choose_decoded_value * S ((S (p * r + e)) * bcf_row_scale_lucas_block_digit_zero_upper_choose) + (C))))))))) -> C = 0

Constructive proof overview

Generated structural guide

The zero-upper-quotient boundary of the Lucas prime block is constructively zero in every positive lower-quotient column.

The unchanged tactic script uses 3 declared prerequisites and contains 25 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

choose_upper_eq_transport Alpha theorem; checked-use authorized LU0010 lucas_prime_block_zero_reassociation LU000K lucas_positive_lower_quotient_digit_coefficient_zero

Direct dependents

LU000M lucas_prime_block_digit_congruence

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

25 script commands · 3 reading checkpoints · 1 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 (2)
lucas prime block zero reassociationlucas positive lower quotient digit coefficient zero

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro r
  3. L3
    intro d
  4. L4
    intro e
  5. L5
    intro C
  6. L6
    intro hdigit
  7. L7
    intro hpositive
  8. L8
    intro hchoose
02Establish hnormalizedL9–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply choose upper eq transport.

  1. L9
    have hnormalized : Choose(d,p · r + e,C)Definitions: Choose
  2. L10
    specialize choose_upper_eq_transport (p * 0 + d)
  3. L11
    specialize choose_upper_eq_transport d
  4. L12
    specialize choose_upper_eq_transport (p * r + e)
  5. L13
    specialize choose_upper_eq_transport C
  6. L14
    apply choose_upper_eq_transport
  7. L15
    apply lucas_prime_block_zero_reassociation
  8. L16
    exact hchoose
  9. L17
    specialize lucas_positive_lower_quotient_digit_coefficient_zero p
  10. L18
    specialize lucas_positive_lower_quotient_digit_coefficient_zero r
03Use earlier factsL19–25

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

  1. L19
    specialize lucas_positive_lower_quotient_digit_coefficient_zero d
  2. L20
    specialize lucas_positive_lower_quotient_digit_coefficient_zero e
  3. L21
    specialize lucas_positive_lower_quotient_digit_coefficient_zero C
  4. L22
    apply lucas_positive_lower_quotient_digit_coefficient_zero
  5. L23
    exact hdigit
  6. L24
    exact hpositive
  7. L25
    exact hnormalized

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro p
  2. 0002intro r
  3. 0003intro d
  4. 0004intro e
  5. 0005intro C
  6. 0006intro hdigit
  7. 0007intro hpositive
  8. 0008intro hchoose
  9. 0009have hnormalized : ((exists bcf_lt_gap_lucas_block_digit_zero_upper_normalized_out_of_range. bcf_lt_gap_lucas_block_digit_zero_upper_normalized_out_of_range + S (d) = p * r + e) /\ C = 0) \/ ((exists bcf_le_gap_lucas_block_digit_zero_upper_normalized_in_range. bcf_le_gap_lucas_block_digit_zero_upper_normalized_in_range + (p * r + e) = d) /\ (exists bcf_row_code_code_lucas_block_digit_zero_upper_normalized bcf_row_code_scale_lucas_block_digit_zero_upper_normalized bcf_row_scale_code_lucas_block_digit_zero_upper_normalized bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized bcf_row_code_lucas_block_digit_zero_upper_normalized bcf_row_scale_lucas_block_digit_zero_upper_normalized. ((forall bcf_row_index_lucas_block_digit_zero_upper_normalized_table. (exists bcf_lt_gap_lucas_block_digit_zero_upper_normalized_table_row_bound. bcf_lt_gap_lucas_block_digit_zero_upper_normalized_table_row_bound + S (bcf_row_index_lucas_block_digit_zero_upper_normalized_table) = S (d)) -> exists bcf_row_code_lucas_block_digit_zero_upper_normalized_table bcf_row_scale_lucas_block_digit_zero_upper_normalized_table. ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_row_code. bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_row_code + S (bcf_row_code_lucas_block_digit_zero_upper_normalized_table) = S ((S (bcf_row_index_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_row_code. bcf_row_code_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_row_code * S ((S (bcf_row_index_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_normalized) + (bcf_row_code_lucas_block_digit_zero_upper_normalized_table))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_row_scale. bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_zero_upper_normalized_table) = S ((S (bcf_row_index_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_row_scale * S ((S (bcf_row_index_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized) + (bcf_row_scale_lucas_block_digit_zero_upper_normalized_table))) /\ ((bcf_row_index_lucas_block_digit_zero_upper_normalized_table = 0 /\ (forall bcf_index_lucas_block_digit_zero_upper_normalized_table_zero_row. (exists bcf_lt_gap_lucas_block_digit_zero_upper_normalized_table_zero_row_bound. bcf_lt_gap_lucas_block_digit_zero_upper_normalized_table_zero_row_bound + S (bcf_index_lucas_block_digit_zero_upper_normalized_table_zero_row) = S (d)) -> exists bcf_value_lucas_block_digit_zero_upper_normalized_table_zero_row. ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_zero_row_entry. bcf_height_lucas_block_digit_zero_upper_normalized_table_zero_row_entry + S (bcf_value_lucas_block_digit_zero_upper_normalized_table_zero_row) = S ((S (bcf_index_lucas_block_digit_zero_upper_normalized_table_zero_row)) * bcf_row_scale_lucas_block_digit_zero_upper_normalized_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_zero_row_entry. bcf_row_code_lucas_block_digit_zero_upper_normalized_table = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_zero_row_entry * S ((S (bcf_index_lucas_block_digit_zero_upper_normalized_table_zero_row)) * bcf_row_scale_lucas_block_digit_zero_upper_normalized_table) + (bcf_value_lucas_block_digit_zero_upper_normalized_table_zero_row))) /\ ((bcf_index_lucas_block_digit_zero_upper_normalized_table_zero_row = 0 /\ bcf_value_lucas_block_digit_zero_upper_normalized_table_zero_row = 1) \/ exists bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_zero_row. bcf_index_lucas_block_digit_zero_upper_normalized_table_zero_row = S bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_zero_row /\ bcf_value_lucas_block_digit_zero_upper_normalized_table_zero_row = 0)))) \/ exists bcf_predecessor_lucas_block_digit_zero_upper_normalized_table bcf_previous_code_lucas_block_digit_zero_upper_normalized_table bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table. bcf_row_index_lucas_block_digit_zero_upper_normalized_table = S bcf_predecessor_lucas_block_digit_zero_upper_normalized_table /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_previous_code. bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_previous_code + S (bcf_previous_code_lucas_block_digit_zero_upper_normalized_table) = S ((S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_previous_code. bcf_row_code_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_previous_code * S ((S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_code_scale_lucas_block_digit_zero_upper_normalized) + (bcf_previous_code_lucas_block_digit_zero_upper_normalized_table))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_previous_scale. bcf_height_lucas_block_digit_zero_upper_normalized_table_decoded_previous_scale + S (bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table) = S ((S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_previous_scale. bcf_row_scale_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_decoded_previous_scale * S ((S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized) + (bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table))) /\ (forall bcf_index_lucas_block_digit_zero_upper_normalized_table_row_step. (exists bcf_lt_gap_lucas_block_digit_zero_upper_normalized_table_row_step_bound. bcf_lt_gap_lucas_block_digit_zero_upper_normalized_table_row_step_bound + S (bcf_index_lucas_block_digit_zero_upper_normalized_table_row_step) = S (d)) -> exists bcf_value_lucas_block_digit_zero_upper_normalized_table_row_step. ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_row_step_entry. bcf_height_lucas_block_digit_zero_upper_normalized_table_row_step_entry + S (bcf_value_lucas_block_digit_zero_upper_normalized_table_row_step) = S ((S (bcf_index_lucas_block_digit_zero_upper_normalized_table_row_step)) * bcf_row_scale_lucas_block_digit_zero_upper_normalized_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_row_step_entry. bcf_row_code_lucas_block_digit_zero_upper_normalized_table = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_row_step_entry * S ((S (bcf_index_lucas_block_digit_zero_upper_normalized_table_row_step)) * bcf_row_scale_lucas_block_digit_zero_upper_normalized_table) + (bcf_value_lucas_block_digit_zero_upper_normalized_table_row_step))) /\ ((bcf_index_lucas_block_digit_zero_upper_normalized_table_row_step = 0 /\ bcf_value_lucas_block_digit_zero_upper_normalized_table_row_step = 1) \/ exists bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_row_step bcf_left_lucas_block_digit_zero_upper_normalized_table_row_step bcf_right_lucas_block_digit_zero_upper_normalized_table_row_step. bcf_index_lucas_block_digit_zero_upper_normalized_table_row_step = S bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_row_step /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_row_step_previous_left. bcf_height_lucas_block_digit_zero_upper_normalized_table_row_step_previous_left + S (bcf_left_lucas_block_digit_zero_upper_normalized_table_row_step) = S ((S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_row_step)) * bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_row_step_previous_left. bcf_previous_code_lucas_block_digit_zero_upper_normalized_table = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_row_step_previous_left * S ((S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_row_step)) * bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table) + (bcf_left_lucas_block_digit_zero_upper_normalized_table_row_step))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_table_row_step_previous_right. bcf_height_lucas_block_digit_zero_upper_normalized_table_row_step_previous_right + S (bcf_right_lucas_block_digit_zero_upper_normalized_table_row_step) = S ((S (S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_row_step))) * bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_table_row_step_previous_right. bcf_previous_code_lucas_block_digit_zero_upper_normalized_table = bcf_quotient_lucas_block_digit_zero_upper_normalized_table_row_step_previous_right * S ((S (S (bcf_predecessor_lucas_block_digit_zero_upper_normalized_table_row_step))) * bcf_previous_scale_lucas_block_digit_zero_upper_normalized_table) + (bcf_right_lucas_block_digit_zero_upper_normalized_table_row_step))) /\ bcf_value_lucas_block_digit_zero_upper_normalized_table_row_step = bcf_left_lucas_block_digit_zero_upper_normalized_table_row_step + bcf_right_lucas_block_digit_zero_upper_normalized_table_row_step))))))))))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_decoded_row_code. bcf_height_lucas_block_digit_zero_upper_normalized_decoded_row_code + S (bcf_row_code_lucas_block_digit_zero_upper_normalized) = S ((S (d)) * bcf_row_code_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_decoded_row_code. bcf_row_code_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_decoded_row_code * S ((S (d)) * bcf_row_code_scale_lucas_block_digit_zero_upper_normalized) + (bcf_row_code_lucas_block_digit_zero_upper_normalized))) /\ ((((exists bcf_height_lucas_block_digit_zero_upper_normalized_decoded_row_scale. bcf_height_lucas_block_digit_zero_upper_normalized_decoded_row_scale + S (bcf_row_scale_lucas_block_digit_zero_upper_normalized) = S ((S (d)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_decoded_row_scale. bcf_row_scale_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_decoded_row_scale * S ((S (d)) * bcf_row_scale_scale_lucas_block_digit_zero_upper_normalized) + (bcf_row_scale_lucas_block_digit_zero_upper_normalized))) /\ (((exists bcf_height_lucas_block_digit_zero_upper_normalized_decoded_value. bcf_height_lucas_block_digit_zero_upper_normalized_decoded_value + S (C) = S ((S (p * r + e)) * bcf_row_scale_lucas_block_digit_zero_upper_normalized)) /\ exists bcf_quotient_lucas_block_digit_zero_upper_normalized_decoded_value. bcf_row_code_lucas_block_digit_zero_upper_normalized = bcf_quotient_lucas_block_digit_zero_upper_normalized_decoded_value * S ((S (p * r + e)) * bcf_row_scale_lucas_block_digit_zero_upper_normalized) + (C))))))))
  10. 0010specialize choose_upper_eq_transport (p * 0 + d)
  11. 0011specialize choose_upper_eq_transport d
  12. 0012specialize choose_upper_eq_transport (p * r + e)
  13. 0013specialize choose_upper_eq_transport C
  14. 0014apply choose_upper_eq_transport
  15. 0015apply lucas_prime_block_zero_reassociation
  16. 0016exact hchoose
  17. 0017specialize lucas_positive_lower_quotient_digit_coefficient_zero p
  18. 0018specialize lucas_positive_lower_quotient_digit_coefficient_zero r
  19. 0019specialize lucas_positive_lower_quotient_digit_coefficient_zero d
  20. 0020specialize lucas_positive_lower_quotient_digit_coefficient_zero e
  21. 0021specialize lucas_positive_lower_quotient_digit_coefficient_zero C
  22. 0022apply lucas_positive_lower_quotient_digit_coefficient_zero
  23. 0023exact hdigit
  24. 0024exact hpositive
  25. 0025exact hnormalized