LU001A · theorem body

lucas_prime_digit_chain_exists

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

Every prime modulus admits arbitrarily long coherent constructive beta-coded base-p digit streams.

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

∀ p. ∀ n. ∀ l. Prime(p) → ∃ x. ∃ y. ∃ z. ∃ m. BetaAt(x,y,0,n) ∧ (∀ k. Lt(k,l) → ∃ i. ∃ j. ∃ u. BetaAt(x,y,k,i) ∧ (BetaAt(x,y,S k,j) ∧ (BetaAt(z,m,k,u) ∧ DivRem(i,p,j,u))))

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 PD0004 Prime PD0007 DivRem PD0013 BetaAt

In local proof propositions

none
Exact expanded first-order statement
forall p n l. ((~(p = 1) /\ forall frm_prime_left_lmd_chain_prime frm_prime_right_lmd_chain_prime. p = frm_prime_left_lmd_chain_prime * frm_prime_right_lmd_chain_prime -> frm_prime_left_lmd_chain_prime = 1 \/ frm_prime_right_lmd_chain_prime = 1)) -> exists qb qc db dc. (((((exists ff_h_lmd_total_result_initial. ff_h_lmd_total_result_initial + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_total_result_initial. qb = ff_q_lmd_total_result_initial * S ((S (0)) * qc) + (n))) /\ forall lmd_index_total_result. (exists lmd_gap_total_result_index. lmd_gap_total_result_index + S (lmd_index_total_result) = (l)) -> exists lmd_current_total_result lmd_successor_total_result lmd_digit_total_result. ((((exists ff_h_lmd_total_result_current. ff_h_lmd_total_result_current + S (lmd_current_total_result) = S ((S (lmd_index_total_result)) * qc)) /\ exists ff_q_lmd_total_result_current. qb = ff_q_lmd_total_result_current * S ((S (lmd_index_total_result)) * qc) + (lmd_current_total_result))) /\ ((((exists ff_h_lmd_total_result_successor. ff_h_lmd_total_result_successor + S (lmd_successor_total_result) = S ((S (S lmd_index_total_result)) * qc)) /\ exists ff_q_lmd_total_result_successor. qb = ff_q_lmd_total_result_successor * S ((S (S lmd_index_total_result)) * qc) + (lmd_successor_total_result))) /\ ((((exists ff_h_lmd_total_result_digit. ff_h_lmd_total_result_digit + S (lmd_digit_total_result) = S ((S (lmd_index_total_result)) * dc)) /\ exists ff_q_lmd_total_result_digit. db = ff_q_lmd_total_result_digit * S ((S (lmd_index_total_result)) * dc) + (lmd_digit_total_result))) /\ ((lmd_current_total_result = (p) * (lmd_successor_total_result) + (lmd_digit_total_result)) /\ (exists lmd_gap_total_result_digit_bound. lmd_gap_total_result_digit_bound + S (lmd_digit_total_result) = (p))))))))

Proof neighborhood

Direct theorem prerequisites

prime_nonzero · Stable closed LU0019 lucas_digit_chain_exists

Direct theorem dependents

LU001N lucas_terminating_prime_digit_chain_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 · 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
lucas digit chain exists
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro l
  4. L4
    intro hprime
02Use earlier factsL5–8

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

  1. L5
    specialize lucas_digit_chain_exists p
  2. L6
    specialize lucas_digit_chain_exists n
  3. L7
    specialize lucas_digit_chain_exists l
  4. L8
    apply lucas_digit_chain_exists
03Fix variables and assumptionsL9–9

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

  1. L9
    intro hzero
04Use earlier factsL10–13

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

  1. L10
    specialize prime_nonzero p
  2. L11
    apply prime_nonzero
  3. L12
    exact hprime
  4. L13
    exact hzero

Library-wide reading audit

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