EL000F

lte_prime_nondivisor_two

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

Two is a nondivisor unit for every prime explicitly different from two.

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. (~((p) = 1) /\ forall pvs_left_odd_prime pvs_right_odd_prime. (p) = pvs_left_odd_prime * pvs_right_odd_prime -> pvs_left_odd_prime = 1 \/ pvs_right_odd_prime = 1) -> ~(p = 2) -> ~(exists olte_factor_two. (2) = (p) * olte_factor_two)

Constructive proof overview

Generated structural guide

Two is a nondivisor unit for every prime explicitly different from two.

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_divisor_of_prime_forces_equality Alpha theorem; checked-use authorized prime_two 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 hp
  3. L3
    intro hne
  4. L4
    intro hdiv
02Use earlier factsL5–11

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

  1. L5
    apply hne
  2. L6
    specialize prime_divisor_of_prime_forces_equality (p)
  3. L7
    specialize prime_divisor_of_prime_forces_equality (2)
  4. L8
    apply prime_divisor_of_prime_forces_equality
  5. L9
    exact hp
  6. L10
    exact prime_two
  7. L11
    exact hdiv

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003intro hne
  4. 0004intro hdiv
  5. 0005apply hne
  6. 0006specialize prime_divisor_of_prime_forces_equality (p)
  7. 0007specialize prime_divisor_of_prime_forces_equality (2)
  8. 0008apply prime_divisor_of_prime_forces_equality
  9. 0009exact hp
  10. 0010exact prime_two
  11. 0011exact hdiv