LU000B

lucas_prime_base_digit_prefix_exists

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Every prime base constructively digitizes an entire finite beta-coded source prefix.

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 b c l. ((~(p = 1) /\ forall frm_prime_left_lucas_prefix_prime frm_prime_right_lucas_prefix_prime. p = frm_prime_left_lucas_prefix_prime * frm_prime_right_lucas_prefix_prime -> frm_prime_left_lucas_prefix_prime = 1 \/ frm_prime_right_lucas_prefix_prime = 1)) -> exists qb qc db dc. (forall fdp_index_lucas_digits. (exists gsp_lt_gap_lucas_digits_index_bound. gsp_lt_gap_lucas_digits_index_bound + S fdp_index_lucas_digits = l) -> exists fdp_value_lucas_digits fdp_quotient_lucas_digits fdp_remainder_lucas_digits. (((exists ff_h_fdp_lucas_digits_source. ff_h_fdp_lucas_digits_source + S (fdp_value_lucas_digits) = S ((S (fdp_index_lucas_digits)) * c)) /\ exists ff_q_fdp_lucas_digits_source. b = ff_q_fdp_lucas_digits_source * S ((S (fdp_index_lucas_digits)) * c) + (fdp_value_lucas_digits))) /\ ((((exists ff_h_fdp_lucas_digits_quotient_entry. ff_h_fdp_lucas_digits_quotient_entry + S (fdp_quotient_lucas_digits) = S ((S (fdp_index_lucas_digits)) * qc)) /\ exists ff_q_fdp_lucas_digits_quotient_entry. qb = ff_q_fdp_lucas_digits_quotient_entry * S ((S (fdp_index_lucas_digits)) * qc) + (fdp_quotient_lucas_digits))) /\ ((((exists ff_h_fdp_lucas_digits_remainder_entry. ff_h_fdp_lucas_digits_remainder_entry + S (fdp_remainder_lucas_digits) = S ((S (fdp_index_lucas_digits)) * dc)) /\ exists ff_q_fdp_lucas_digits_remainder_entry. db = ff_q_fdp_lucas_digits_remainder_entry * S ((S (fdp_index_lucas_digits)) * dc) + (fdp_remainder_lucas_digits))) /\ (fdp_value_lucas_digits = p * fdp_quotient_lucas_digits + fdp_remainder_lucas_digits /\ (exists gsp_lt_gap_lucas_digits_remainder_bound. gsp_lt_gap_lucas_digits_remainder_bound + S fdp_remainder_lucas_digits = p)))))

Constructive proof overview

Generated structural guide

Every prime base constructively digitizes an entire finite beta-coded source prefix.

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

Proof neighborhood

Direct dependencies

prime_nonzero Stable theorem; checked-use authorized LU000A lucas_base_p_digit_prefix_exists

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

15 script commands · 4 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 base p digit prefix exists
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro hprime
02Use earlier factsL6–10

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

  1. L6
    specialize lucas_base_p_digit_prefix_exists p
  2. L7
    specialize lucas_base_p_digit_prefix_exists b
  3. L8
    specialize lucas_base_p_digit_prefix_exists c
  4. L9
    specialize lucas_base_p_digit_prefix_exists l
  5. L10
    apply lucas_base_p_digit_prefix_exists
03Fix variables and assumptionsL11–11

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

  1. L11
    intro hzero
04Use earlier factsL12–15

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

  1. L12
    specialize prime_nonzero p
  2. L13
    apply prime_nonzero
  3. L14
    exact hprime
  4. L15
    exact hzero

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hprime
  6. 0006specialize lucas_base_p_digit_prefix_exists p
  7. 0007specialize lucas_base_p_digit_prefix_exists b
  8. 0008specialize lucas_base_p_digit_prefix_exists c
  9. 0009specialize lucas_base_p_digit_prefix_exists l
  10. 0010apply lucas_base_p_digit_prefix_exists
  11. 0011intro hzero
  12. 0012specialize prime_nonzero p
  13. 0013apply prime_nonzero
  14. 0014exact hprime
  15. 0015exact hzero