SK0002

squarefree_excludes_prime_square

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

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

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 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 21 exact native proof lines.

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

Proof neighborhood

Direct dependencies

divisor_le_nonzero Stable theorem; checked-use authorized multiple_trans Stable 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

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.

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 exact 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