LU000B · theorem body

lucas_prime_base_digit_prefix_exists

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

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.

Statement with defined notation

∀ p. ∀ b. ∀ c. ∀ l. Prime(p) → ∃ x. ∃ y. ∃ z. ∃ n. DivisionPrefix(p,b,c,x,y,z,n,l)

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

PD0004 Prime PD0040 DivisionPrefix

In local proof propositions

none
Exact expanded first-order 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)))))

Proof neighborhood

Direct theorem prerequisites

prime_nonzero · Stable closed LU000A lucas_base_p_digit_prefix_exists

Direct theorem dependents

none

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

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.

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 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 defined 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