EL000D

lte_nondivisor_product_right

Nondivisibility of a product excludes divisibility of its right factor.

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. ∀ a. ∀ b. ¬Dvd(p,a · b) → ¬Dvd(p,b)

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

Definition DAG

Dvd(d,n) · 2

Actual proof prerequisites

multiple_mul_left · checked external prerequisite
Original expanded first-order statement 
forall p a b. ~(exists olte_factor_product. (a * b) = (p) * olte_factor_product) -> ~(exists olte_factor_right. (b) = (p) * olte_factor_right)

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 assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hnot
  5. L5
    intro hdiv
02Use earlier factsL6–11

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

  1. L6
    apply hnot
  2. L7
    specialize multiple_mul_left (p)
  3. L8
    specialize multiple_mul_left (b)
  4. L9
    specialize multiple_mul_left (a)
  5. L10
    apply multiple_mul_left
  6. L11
    exact hdiv

Library-wide reading audit

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