SK0002

squarefree_excludes_prime_square

A squared prime divisor cannot evade the bounded squarefree definition: its base prime is itself a divisor and is at most the positive input.

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

∀ n. ∀ p. Squarefree(n) → Prime(p) → ¬Dvd(p · p,n)

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

Definition DAG

Squarefree(n) · 1Dvd(d,n) · 1Prime(p) · 1

Actual proof prerequisites

divisor_le_nonzero · checked external prerequisitemultiple_trans · checked external prerequisite
Original expanded first-order statement 
forall n p. (((~((n) = 0)) /\ (forall sfd_prime_exclusion_squarefree. (~((sfd_prime_exclusion_squarefree) = 1) /\ forall pvs_left_exclusion_squarefreedomain pvs_right_exclusion_squarefreedomain. (sfd_prime_exclusion_squarefree) = pvs_left_exclusion_squarefreedomain * pvs_right_exclusion_squarefreedomain -> pvs_left_exclusion_squarefreedomain = 1 \/ pvs_right_exclusion_squarefreedomain = 1) -> (exists pvs_le_gap_exclusion_squarefreebound. pvs_le_gap_exclusion_squarefreebound + (sfd_prime_exclusion_squarefree) = (n)) -> ~(exists pvs_factor_exclusion_squarefreesquare. (n) = (sfd_prime_exclusion_squarefree * sfd_prime_exclusion_squarefree) * pvs_factor_exclusion_squarefreesquare)))) -> (~((p) = 1) /\ forall pvs_left_exclusion_prime pvs_right_exclusion_prime. (p) = pvs_left_exclusion_prime * pvs_right_exclusion_prime -> pvs_left_exclusion_prime = 1 \/ pvs_right_exclusion_prime = 1) -> (exists pvs_factor_exclusion_divisor. (n) = (p * p) * pvs_factor_exclusion_divisor) -> false

Complete tactic proof in conservative notation

All 21 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

21 script commands · 7 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 n
  2. L2
    intro p
  3. L3
    intro hsf
  4. L4
    intro hp
  5. L5
    intro hdiv
02Separate the logical casesL6–6

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

  1. L6
    cases hsf
03Use earlier factsL7–16

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

  1. L7
    specialize hsf_right (p)
  2. L8
    apply hsf_right
  3. L9
    exact hp
  4. L10
    specialize divisor_le_nonzero (p)
  5. L11
    specialize divisor_le_nonzero (n)
  6. L12
    apply divisor_le_nonzero
  7. L13
    exact hsf_left
  8. L14
    specialize multiple_trans (p * p)
  9. L15
    specialize multiple_trans (p)
  10. L16
    specialize multiple_trans (n)
04Use earlier factsL17–18

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

  1. L17
    apply multiple_trans
  2. L18
    exact hdiv
05Construct an explicit witnessL19–19

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

  1. L19
    exists p
06Calculate and transport equalitiesL20–20

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

  1. L20
    refl
07Use earlier factsL21–21

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

  1. L21
    exact hdiv

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro n
  2. 0002intro p
  3. 0003intro hsf
  4. 0004intro hp
  5. 0005intro hdiv
  6. 0006cases hsf
  7. 0007specialize hsf_right (p)
  8. 0008apply hsf_right
  9. 0009exact hp
  10. 0010specialize divisor_le_nonzero (p)
  11. 0011specialize divisor_le_nonzero (n)
  12. 0012apply divisor_le_nonzero
  13. 0013exact hsf_left
  14. 0014specialize multiple_trans (p * p)
  15. 0015specialize multiple_trans (p)
  16. 0016specialize multiple_trans (n)
  17. 0017apply multiple_trans
  18. 0018exact hdiv
  19. 0019exists p
  20. 0020refl
  21. 0021exact hdiv