SK000D

squarefree_one

The unit is squarefree under the exact positive, prime-square-free definition.

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

Squarefree(1)

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

Definition DAG

Squarefree(n) · 1

Actual proof prerequisites

divisor_one · checked external prerequisitemultiple_trans · checked external prerequisite
Original expanded first-order statement 
((~((1) = 0)) /\ (forall sfd_prime_one. (~((sfd_prime_one) = 1) /\ forall pvs_left_onedomain pvs_right_onedomain. (sfd_prime_one) = pvs_left_onedomain * pvs_right_onedomain -> pvs_left_onedomain = 1 \/ pvs_right_onedomain = 1) -> (exists pvs_le_gap_onebound. pvs_le_gap_onebound + (sfd_prime_one) = (1)) -> ~(exists pvs_factor_onesquare. (1) = (sfd_prime_one * sfd_prime_one) * pvs_factor_onesquare)))

Complete tactic proof in conservative notation

All 19 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

19 script commands · 8 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.

01Separate the logical casesL1–1

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

  1. L1
    split
02Fix variables and assumptionsL2–2

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

  1. L2
    intro hz
03Use earlier factsL3–4

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

  1. L3
    apply PA1
  2. L4
    exact hz
04Fix variables and assumptionsL5–8

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

  1. L5
    intro p
  2. L6
    intro hp
  3. L7
    intro hle
  4. L8
    intro hdiv
05Separate the logical casesL9–9

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

  1. L9
    cases hp
06Use earlier factsL10–17

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

  1. L10
    apply hp_left
  2. L11
    specialize divisor_one (p)
  3. L12
    apply divisor_one
  4. L13
    specialize multiple_trans (p * p)
  5. L14
    specialize multiple_trans (p)
  6. L15
    specialize multiple_trans (1)
  7. L16
    apply multiple_trans
  8. L17
    exact hdiv
07Construct an explicit witnessL18–18

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

  1. L18
    exists p
08Calculate and transport equalitiesL19–19

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

  1. L19
    refl

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001split
  2. 0002intro hz
  3. 0003apply PA1
  4. 0004exact hz
  5. 0005intro p
  6. 0006intro hp
  7. 0007intro hle
  8. 0008intro hdiv
  9. 0009cases hp
  10. 0010apply hp_left
  11. 0011specialize divisor_one (p)
  12. 0012apply divisor_one
  13. 0013specialize multiple_trans (p * p)
  14. 0014specialize multiple_trans (p)
  15. 0015specialize multiple_trans (1)
  16. 0016apply multiple_trans
  17. 0017exact hdiv
  18. 0018exists p
  19. 0019refl