SK0001

divides_square_of_divides

Squaring an actual divisor witness produces an actual squared divisor witness.

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.

For n>1 the exponent gcd and a real beta table classify and witness all positive root degrees. The unit n=1 has a separate uniform certificate for every positive degree. Zero is excluded. NaturalSquarefreeDecomposition is deliberately distinct from the unrelated polynomial definition.

Exact theorem in conservative defined notation

∀ a. ∀ b. Dvd(a,b) → Dvd(a · a,b · b)

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

Definition DAG

Dvd(d,n) · 2

Actual proof prerequisites

four_square_product_square · checked external prerequisite
Original expanded first-order statement 
forall a b. (exists pvs_factor_square_source. (b) = (a) * pvs_factor_square_source) -> (exists pvs_factor_square_result. (b * b) = (a * a) * pvs_factor_square_result)

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 · 5 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 a
  2. L2
    intro b
  3. L3
    intro hdiv
02Separate the logical casesL4–4

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases hdiv
03Construct an explicit witnessL5–5

Supply the displayed value, then prove that it has the required property.

  1. L5
    exists x * x
04Calculate and transport equalitiesL6–7

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

  1. L6
    rewrite hdiv_witness
  2. L7
    rewrite hdiv_witness
05Use earlier factsL8–8

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

  1. L8
    apply four_square_product_square

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro hdiv
  4. 0004cases hdiv
  5. 0005exists x * x
  6. 0006rewrite hdiv_witness
  7. 0007rewrite hdiv_witness
  8. 0008apply four_square_product_square