EL0023

lte_exceeds_two_not_two

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

The exact p>2 guard excludes the binary prime without an implicit case extension.

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. (exists olte_gap_above_two. olte_gap_above_two + S (2) = (p)) -> ~(p = 2)

Constructive proof overview

Generated structural guide

The exact p>2 guard excludes the binary prime without an implicit case extension.

The unchanged tactic script uses 1 declared prerequisite and contains 7 exact native proof lines.

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

Proof neighborhood

Direct dependencies

lt_irrefl_expanded Stable theorem; checked-use authorized

Direct dependents

EL0024 odd_prime_lifting_the_exponent

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

7 script commands · 3 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 assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro hgt
  3. L3
    intro heq
02Calculate and transport equalitiesL4–4

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L4
    rewrite heq at hgt
03Use earlier factsL5–7

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

  1. L5
    specialize lt_irrefl_expanded (2)
  2. L6
    apply lt_irrefl_expanded
  3. L7
    exact hgt

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro p
  2. 0002intro hgt
  3. 0003intro heq
  4. 0004rewrite heq at hgt
  5. 0005specialize lt_irrefl_expanded (2)
  6. 0006apply lt_irrefl_expanded
  7. 0007exact hgt