SK0004

squarefree_squared_divisor_is_one

Every squared divisor of a positive squarefree number has root one, not merely prime roots.

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. ∀ a. Squarefree(n) → Dvd(a · a,n) → a = 1

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

Definition DAG

Squarefree(n) · 1Dvd(d,n) · 2Prime(p) · 1

Actual proof prerequisites

eq_decidable · checked external prerequisiteprime_divisor_exists · checked external prerequisitesquarefree_excludes_prime_squaremultiple_trans · checked external prerequisitedivides_square_of_dividesmul_zero_left · checked external prerequisite
Original expanded first-order statement 
forall n a. (((~((n) = 0)) /\ (forall sfd_prime_squared_divisor_sf. (~((sfd_prime_squared_divisor_sf) = 1) /\ forall pvs_left_squared_divisor_sfdomain pvs_right_squared_divisor_sfdomain. (sfd_prime_squared_divisor_sf) = pvs_left_squared_divisor_sfdomain * pvs_right_squared_divisor_sfdomain -> pvs_left_squared_divisor_sfdomain = 1 \/ pvs_right_squared_divisor_sfdomain = 1) -> (exists pvs_le_gap_squared_divisor_sfbound. pvs_le_gap_squared_divisor_sfbound + (sfd_prime_squared_divisor_sf) = (n)) -> ~(exists pvs_factor_squared_divisor_sfsquare. (n) = (sfd_prime_squared_divisor_sf * sfd_prime_squared_divisor_sf) * pvs_factor_squared_divisor_sfsquare)))) -> (exists pvs_factor_squared_divisor_input. (n) = (a * a) * pvs_factor_squared_divisor_input) -> a = 1

Complete tactic proof in conservative notation

All 46 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

46 script commands · 15 reading checkpoints · 3 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.

Named ingredients (2)
squarefree excludes prime squaredivides square of divides
01Fix variables and assumptionsL1–4

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

  1. L1
    intro n
  2. L2
    intro a
  3. L3
    intro hsf
  4. L4
    intro hdiv
02Establish hzeroL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eq decidable.

  1. L5
    have hzero : a = 0 \/ ~(a = 0)
  2. L6
    specialize eq_decidable (a)
  3. L7
    specialize eq_decidable (0)
  4. L8
    apply eq_decidable
03Separate the logical casesL9–11

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

  1. L9
    cases hzero
  2. L10
    exfalso
  3. L11
    cases hsf
04Use earlier factsL12–12

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

  1. L12
    apply hsf_left
05Separate the logical casesL13–13

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

  1. L13
    cases hdiv
06Calculate and transport equalitiesL14–14

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

  1. L14
    trans (a * a) * x
07Use earlier factsL15–15

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

  1. L15
    exact hdiv_witness
08Calculate and transport equalitiesL16–18

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

  1. L16
    rewrite hzero_left
  2. L17
    rewrite hzero_left
  3. L18
    simp [mul_zero_left]
09Establish honeL19–22

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eq decidable.

  1. L19
    have hone : a = 1 \/ ~(a = 1)
  2. L20
    specialize eq_decidable (a)
  3. L21
    specialize eq_decidable (1)
  4. L22
    apply eq_decidable
10Separate the logical casesL23–23

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

  1. L23
    cases hone
11Use earlier factsL24–24

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

  1. L24
    exact hone_left
12Establish hpL25–29

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime divisor exists.

  1. L25
    have hp : ∃ p. Prime(p) ∧ Dvd(p,a)Definitions: Prime(p)Dvd(p,a)Original native command in the exact edition
  2. L26
    specialize prime_divisor_exists (a)
  3. L27
    apply prime_divisor_exists
  4. L28
    exact hzero_right
  5. L29
    exact hone_right
13Separate the logical casesL30–32

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

  1. L30
    cases hp
  2. L31
    cases hp_witness
  3. L32
    exfalso
14Use earlier factsL33–42

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

  1. L33
    specialize squarefree_excludes_prime_square (n)
  2. L34
    specialize squarefree_excludes_prime_square (x)
  3. L35
    apply squarefree_excludes_prime_square
  4. L36
    exact hsf
  5. L37
    exact hp_witness_left
  6. L38
    specialize multiple_trans (a * a)
  7. L39
    specialize multiple_trans (x * x)
  8. L40
    specialize multiple_trans (n)
  9. L41
    apply multiple_trans
  10. L42
    exact hdiv
15Use earlier factsL43–46

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

  1. L43
    specialize divides_square_of_divides (x)
  2. L44
    specialize divides_square_of_divides (a)
  3. L45
    apply divides_square_of_divides
  4. L46
    exact hp_witness_right

Library-wide reading audit

Original defined command ledger · 46 lines
  1. 0001intro n
  2. 0002intro a
  3. 0003intro hsf
  4. 0004intro hdiv
  5. 0005have hzero : a = 0 \/ ~(a = 0)
  6. 0006specialize eq_decidable (a)
  7. 0007specialize eq_decidable (0)
  8. 0008apply eq_decidable
  9. 0009cases hzero
  10. 0010exfalso
  11. 0011cases hsf
  12. 0012apply hsf_left
  13. 0013cases hdiv
  14. 0014trans (a * a) * x
  15. 0015exact hdiv_witness
  16. 0016rewrite hzero_left
  17. 0017rewrite hzero_left
  18. 0018simp [mul_zero_left]
  19. 0019have hone : a = 1 \/ ~(a = 1)
  20. 0020specialize eq_decidable (a)
  21. 0021specialize eq_decidable (1)
  22. 0022apply eq_decidable
  23. 0023cases hone
  24. 0024exact hone_left
  25. 0025have hp : ∃ p. Prime(p) ∧ Dvd(p,a)
  26. 0026specialize prime_divisor_exists (a)
  27. 0027apply prime_divisor_exists
  28. 0028exact hzero_right
  29. 0029exact hone_right
  30. 0030cases hp
  31. 0031cases hp_witness
  32. 0032exfalso
  33. 0033specialize squarefree_excludes_prime_square (n)
  34. 0034specialize squarefree_excludes_prime_square (x)
  35. 0035apply squarefree_excludes_prime_square
  36. 0036exact hsf
  37. 0037exact hp_witness_left
  38. 0038specialize multiple_trans (a * a)
  39. 0039specialize multiple_trans (x * x)
  40. 0040specialize multiple_trans (n)
  41. 0041apply multiple_trans
  42. 0042exact hdiv
  43. 0043specialize divides_square_of_divides (x)
  44. 0044specialize divides_square_of_divides (a)
  45. 0045apply divides_square_of_divides
  46. 0046exact hp_witness_right