EL0023

lte_exceeds_two_not_two

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

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

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.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ p. Lt(2,p) → ¬p = 2

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

lt_irrefl_expanded · checked external prerequisite
Original expanded first-order statement 
forall p. (exists olte_gap_above_two. olte_gap_above_two + S (2) = (p)) -> ~(p = 2)

Complete tactic proof in conservative notation

All 7 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

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

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 defined 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