EL000F

lte_prime_nondivisor_two

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

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. ¬p = 1 ∧ (∀ x. ∀ y. p = x · y → x = 1 ∨ y = 1) → ¬p = 2 → ¬Dvd(p,2)

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

Definition DAG

Dvd(d,n) · 1

Actual proof prerequisites

prime_divisor_of_prime_forces_equality · checked external prerequisiteprime_two · checked external prerequisite
Original expanded first-order 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)

Complete tactic proof in conservative notation

All 11 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

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.

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 assumptionsL1–4

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 defined 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