LU000X

lucas_prime_plus_index_nonzero

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

Adding a natural index to a nonzero prime never produces 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 j. ((~(p = 1) /\ forall frm_prime_left_lucas_convolution_prime frm_prime_right_lucas_convolution_prime. p = frm_prime_left_lucas_convolution_prime * frm_prime_right_lucas_convolution_prime -> frm_prime_left_lucas_convolution_prime = 1 \/ frm_prime_right_lucas_convolution_prime = 1)) -> ~(p + j = 0)

Constructive proof overview

Generated structural guide

Adding a natural index to a nonzero prime never produces zero.

The unchanged tactic script uses 2 declared prerequisites and contains 11 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

prime_nonzero Stable theorem; checked-use authorized add_eq_zero_left Stable theorem; checked-use authorized

Direct dependents

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

11 script commands · 2 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro j
  3. L3
    intro hprime
  4. L4
    intro hzero
02Use earlier factsL5–11

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

  1. L5
    specialize prime_nonzero p
  2. L6
    apply prime_nonzero
  3. L7
    exact hprime
  4. L8
    specialize add_eq_zero_left p
  5. L9
    specialize add_eq_zero_left j
  6. L10
    apply add_eq_zero_left
  7. L11
    exact hzero

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro p
  2. 0002intro j
  3. 0003intro hprime
  4. 0004intro hzero
  5. 0005specialize prime_nonzero p
  6. 0006apply prime_nonzero
  7. 0007exact hprime
  8. 0008specialize add_eq_zero_left p
  9. 0009specialize add_eq_zero_left j
  10. 0010apply add_eq_zero_left
  11. 0011exact hzero