LU001M

lucas_prime_digit_chain_terminal_zero

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

Every prime-base digit trace longer than its original natural has an actual terminal quotient equal to zero.

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 n qb qc db dc l q. ((~(p = 1) /\ forall frm_prime_left_lmd_terminal_bound_prime frm_prime_right_lmd_terminal_bound_prime. p = frm_prime_left_lmd_terminal_bound_prime * frm_prime_right_lmd_terminal_bound_prime -> frm_prime_left_lmd_terminal_bound_prime = 1 \/ frm_prime_right_lmd_terminal_bound_prime = 1)) -> (exists lmd_gap_terminal_length. lmd_gap_terminal_length + S (n) = (l)) -> (((((exists ff_h_lmd_zero_terminal_chain_initial. ff_h_lmd_zero_terminal_chain_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_zero_terminal_chain_initial. qb = ff_q_lmd_zero_terminal_chain_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_zero_terminal_chain. (exists lmd_gap_zero_terminal_chain_index. lmd_gap_zero_terminal_chain_index + S (lmd_index_zero_terminal_chain) = (l)) -> exists lmd_current_zero_terminal_chain lmd_successor_zero_terminal_chain lmd_digit_zero_terminal_chain. ((((exists ff_h_lmd_zero_terminal_chain_current. ff_h_lmd_zero_terminal_chain_current + S (lmd_current_zero_terminal_chain) = S ((S (lmd_index_zero_terminal_chain)) * qc)) /\ exists ff_q_lmd_zero_terminal_chain_current. qb = ff_q_lmd_zero_terminal_chain_current * S ((S (lmd_index_zero_terminal_chain)) * qc) + (lmd_current_zero_terminal_chain))) /\ ((((exists ff_h_lmd_zero_terminal_chain_successor. ff_h_lmd_zero_terminal_chain_successor + S (lmd_successor_zero_terminal_chain) = S ((S (S lmd_index_zero_terminal_chain)) * qc)) /\ exists ff_q_lmd_zero_terminal_chain_successor. qb = ff_q_lmd_zero_terminal_chain_successor * S ((S (S lmd_index_zero_terminal_chain)) * qc) + (lmd_successor_zero_terminal_chain))) /\ ((((exists ff_h_lmd_zero_terminal_chain_digit. ff_h_lmd_zero_terminal_chain_digit + S (lmd_digit_zero_terminal_chain) = S ((S (lmd_index_zero_terminal_chain)) * dc)) /\ exists ff_q_lmd_zero_terminal_chain_digit. db = ff_q_lmd_zero_terminal_chain_digit * S ((S (lmd_index_zero_terminal_chain)) * dc) + (lmd_digit_zero_terminal_chain))) /\ ((lmd_current_zero_terminal_chain = (p) * (lmd_successor_zero_terminal_chain) + (lmd_digit_zero_terminal_chain)) /\ (exists lmd_gap_zero_terminal_chain_digit_bound. lmd_gap_zero_terminal_chain_digit_bound + S (lmd_digit_zero_terminal_chain) = (p)))))))) -> (((exists ff_h_lmd_zero_terminal_entry. ff_h_lmd_zero_terminal_entry + S (q) = S ((S (l)) * qc)) /\ exists ff_q_lmd_zero_terminal_entry. qb = ff_q_lmd_zero_terminal_entry * S ((S (l)) * qc) + (q))) -> q = 0

Constructive proof overview

Generated structural guide

Every prime-base digit trace longer than its original natural has an actual terminal quotient equal to zero.

The unchanged tactic script uses 6 declared prerequisites and contains 48 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

eq_decidable Stable theorem; checked-use authorized le_refl Stable theorem; checked-use authorized LU001L lucas_prime_digit_chain_nonzero_index_bound le_add_left Stable theorem; checked-use authorized le_trans Stable theorem; checked-use authorized lt_not_le Stable theorem; checked-use authorized

Direct dependents

LU001N lucas_terminating_prime_digit_chain_exists

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

48 script commands · 10 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.

Named ingredients (1)
lucas prime digit chain nonzero index bound
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro qb
  4. L4
    intro qc
  5. L5
    intro db
  6. L6
    intro dc
  7. L7
    intro l
  8. L8
    intro q
  9. L9
    intro hprime
  10. L10
    intro hlength
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hchain
  2. L12
    intro hentry
03Use earlier factsL13–14

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

  1. L13
    specialize eq_decidable q
  2. L14
    specialize eq_decidable 0
04Separate the logical casesL15–15

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

  1. L15
    cases eq_decidable
05Use earlier factsL16–16

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

  1. L16
    exact eq_decidable_left
06Separate the logical casesL17–17

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

  1. L17
    exfalso
07Establish hboundL18–27

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

  1. L18
    have hbound : exists gap. gap + (q + l) = n
  2. L19
    specialize lucas_prime_digit_chain_nonzero_index_bound l
  3. L20
    specialize lucas_prime_digit_chain_nonzero_index_bound p
  4. L21
    specialize lucas_prime_digit_chain_nonzero_index_bound n
  5. L22
    specialize lucas_prime_digit_chain_nonzero_index_bound qb
  6. L23
    specialize lucas_prime_digit_chain_nonzero_index_bound qc
  7. L24
    specialize lucas_prime_digit_chain_nonzero_index_bound db
  8. L25
    specialize lucas_prime_digit_chain_nonzero_index_bound dc
  9. L26
    specialize lucas_prime_digit_chain_nonzero_index_bound l
  10. L27
    specialize lucas_prime_digit_chain_nonzero_index_bound q
08Use earlier factsL28–34

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

  1. L28
    apply lucas_prime_digit_chain_nonzero_index_bound
  2. L29
    specialize le_refl l
  3. L30
    exact le_refl
  4. L31
    exact hprime
  5. L32
    exact hchain
  6. L33
    exact hentry
  7. L34
    exact eq_decidable_right
09Establish hreverseL35–44

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

  1. L35
    have hreverse : exists gap. gap + l = n
  2. L36
    specialize le_trans l
  3. L37
    specialize le_trans (q + l)
  4. L38
    specialize le_trans n
  5. L39
    apply le_trans
  6. L40
    specialize le_add_left l
  7. L41
    specialize le_add_left q
  8. L42
    exact le_add_left
  9. L43
    exact hbound
  10. L44
    specialize lt_not_le n
10Use earlier factsL45–48

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

  1. L45
    specialize lt_not_le l
  2. L46
    apply lt_not_le
  3. L47
    exact hlength
  4. L48
    exact hreverse

Library-wide reading audit

Original exact command ledger · 48 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro qb
  4. 0004intro qc
  5. 0005intro db
  6. 0006intro dc
  7. 0007intro l
  8. 0008intro q
  9. 0009intro hprime
  10. 0010intro hlength
  11. 0011intro hchain
  12. 0012intro hentry
  13. 0013specialize eq_decidable q
  14. 0014specialize eq_decidable 0
  15. 0015cases eq_decidable
  16. 0016exact eq_decidable_left
  17. 0017exfalso
  18. 0018have hbound : exists gap. gap + (q + l) = n
  19. 0019specialize lucas_prime_digit_chain_nonzero_index_bound l
  20. 0020specialize lucas_prime_digit_chain_nonzero_index_bound p
  21. 0021specialize lucas_prime_digit_chain_nonzero_index_bound n
  22. 0022specialize lucas_prime_digit_chain_nonzero_index_bound qb
  23. 0023specialize lucas_prime_digit_chain_nonzero_index_bound qc
  24. 0024specialize lucas_prime_digit_chain_nonzero_index_bound db
  25. 0025specialize lucas_prime_digit_chain_nonzero_index_bound dc
  26. 0026specialize lucas_prime_digit_chain_nonzero_index_bound l
  27. 0027specialize lucas_prime_digit_chain_nonzero_index_bound q
  28. 0028apply lucas_prime_digit_chain_nonzero_index_bound
  29. 0029specialize le_refl l
  30. 0030exact le_refl
  31. 0031exact hprime
  32. 0032exact hchain
  33. 0033exact hentry
  34. 0034exact eq_decidable_right
  35. 0035have hreverse : exists gap. gap + l = n
  36. 0036specialize le_trans l
  37. 0037specialize le_trans (q + l)
  38. 0038specialize le_trans n
  39. 0039apply le_trans
  40. 0040specialize le_add_left l
  41. 0041specialize le_add_left q
  42. 0042exact le_add_left
  43. 0043exact hbound
  44. 0044specialize lt_not_le n
  45. 0045specialize lt_not_le l
  46. 0046apply lt_not_le
  47. 0047exact hlength
  48. 0048exact hreverse