EL000B

lte_nondivisor_nonzero

A value not divisible by p is nonzero, for every p including zero.

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. ∀ x. ¬Dvd(p,x) → ¬x = 0

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

Definition DAG

Dvd(d,n) · 1

Actual proof prerequisites

multiple_zero · checked external prerequisite
Original expanded first-order statement 
forall p x. ~(exists olte_factor_nonzero. (x) = (p) * olte_factor_nonzero) -> ~(x = 0)

Complete tactic proof in conservative notation

All 8 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

8 script commands · 4 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 x
  3. L3
    intro hnot
  4. L4
    intro hzero
02Use earlier factsL5–5

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

  1. L5
    apply hnot
03Calculate and transport equalitiesL6–6

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

  1. L6
    rewrite hzero
04Use earlier factsL7–8

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

  1. L7
    specialize multiple_zero (p)
  2. L8
    apply multiple_zero

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro p
  2. 0002intro x
  3. 0003intro hnot
  4. 0004intro hzero
  5. 0005apply hnot
  6. 0006rewrite hzero
  7. 0007specialize multiple_zero (p)
  8. 0008apply multiple_zero