SK0001

divides_square_of_divides

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 8 exact native proof lines.

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

Proof neighborhood

Direct dependencies

four_square_product_square Alpha theorem; checked-use authorized

Direct dependents

SK0004 squarefree_squared_divisor_is_one

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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