SK0010

squarefree_decomposition_exists_unique

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

For every positive n, construct n=r*s² with squarefree r and prove every other natural such pair is exactly (r,s), including n=1.

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. ~(n = 0) -> exists r s. (((((~((r) = 0)) /\ (forall sfd_prime_canonical_squarefreekernel. (~((sfd_prime_canonical_squarefreekernel) = 1) /\ forall pvs_left_canonical_squarefreekerneldomain pvs_right_canonical_squarefreekerneldomain. (sfd_prime_canonical_squarefreekernel) = pvs_left_canonical_squarefreekerneldomain * pvs_right_canonical_squarefreekerneldomain -> pvs_left_canonical_squarefreekerneldomain = 1 \/ pvs_right_canonical_squarefreekerneldomain = 1) -> (exists pvs_le_gap_canonical_squarefreekernelbound. pvs_le_gap_canonical_squarefreekernelbound + (sfd_prime_canonical_squarefreekernel) = (r)) -> ~(exists pvs_factor_canonical_squarefreekernelsquare. (r) = (sfd_prime_canonical_squarefreekernel * sfd_prime_canonical_squarefreekernel) * pvs_factor_canonical_squarefreekernelsquare)))) /\ ((n) = (r) * ((s) * (s))))) /\ forall u v. (((((~((u) = 0)) /\ (forall sfd_prime_other_squarefreekernel. (~((sfd_prime_other_squarefreekernel) = 1) /\ forall pvs_left_other_squarefreekerneldomain pvs_right_other_squarefreekerneldomain. (sfd_prime_other_squarefreekernel) = pvs_left_other_squarefreekerneldomain * pvs_right_other_squarefreekerneldomain -> pvs_left_other_squarefreekerneldomain = 1 \/ pvs_right_other_squarefreekerneldomain = 1) -> (exists pvs_le_gap_other_squarefreekernelbound. pvs_le_gap_other_squarefreekernelbound + (sfd_prime_other_squarefreekernel) = (u)) -> ~(exists pvs_factor_other_squarefreekernelsquare. (u) = (sfd_prime_other_squarefreekernel * sfd_prime_other_squarefreekernel) * pvs_factor_other_squarefreekernelsquare)))) /\ ((n) = (u) * ((v) * (v))))) -> u = r /\ v = s

Constructive proof overview

Generated structural guide

For every positive n, construct n=r*s² with squarefree r and prove every other natural such pair is exactly (r,s), including n=1.

The unchanged tactic script uses 2 declared prerequisites and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

SK000C squarefree_decomposition_exists SK000F squarefree_decomposition_functional

Direct dependents

SK0035 positive_squarefree_kernel_and_power_profile

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

28 script commands · 10 reading checkpoints · 1 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.

Named ingredients (2)
squarefree decomposition existssquarefree decomposition functional

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hn
02Establish hexistsL3–6

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

  1. L3
    have hexists : ∃ r. ∃ s. NaturalSquarefreeDecomposition(n,r,s)Definitions: NaturalSquarefreeDecomposition
  2. L4
    specialize squarefree_decomposition_exists (n)
  3. L5
    apply squarefree_decomposition_exists
  4. L6
    exact hn
03Separate the logical casesL7–8

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

  1. L7
    cases hexists
  2. L8
    cases hexists_witness
04Construct an explicit witnessL9–10

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

  1. L9
    exists x
  2. L10
    exists x1
05Separate the logical casesL11–11

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

  1. L11
    split
06Use earlier factsL12–12

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

  1. L12
    exact hexists_witness_witness
07Fix variables and assumptionsL13–15

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

  1. L13
    intro u
  2. L14
    intro v
  3. L15
    intro hother
08Separate the logical casesL16–17

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

  1. L16
    cases hexists_witness_witness
  2. L17
    cases hother
09Use earlier factsL18–27

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

  1. L18
    specialize squarefree_decomposition_functional (n)
  2. L19
    specialize squarefree_decomposition_functional (x)
  3. L20
    specialize squarefree_decomposition_functional (x1)
  4. L21
    specialize squarefree_decomposition_functional (u)
  5. L22
    specialize squarefree_decomposition_functional (v)
  6. L23
    apply squarefree_decomposition_functional
  7. L24
    exact hn
  8. L25
    exact hexists_witness_witness_left
  9. L26
    exact hother_left
  10. L27
    exact hexists_witness_witness_right
10Use earlier factsL28–28

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

  1. L28
    exact hother_right

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003have hexists : exists r s. (((((~((r) = 0)) /\ (forall sfd_prime_canonical_existskernel. (~((sfd_prime_canonical_existskernel) = 1) /\ forall pvs_left_canonical_existskerneldomain pvs_right_canonical_existskerneldomain. (sfd_prime_canonical_existskernel) = pvs_left_canonical_existskerneldomain * pvs_right_canonical_existskerneldomain -> pvs_left_canonical_existskerneldomain = 1 \/ pvs_right_canonical_existskerneldomain = 1) -> (exists pvs_le_gap_canonical_existskernelbound. pvs_le_gap_canonical_existskernelbound + (sfd_prime_canonical_existskernel) = (r)) -> ~(exists pvs_factor_canonical_existskernelsquare. (r) = (sfd_prime_canonical_existskernel * sfd_prime_canonical_existskernel) * pvs_factor_canonical_existskernelsquare)))) /\ ((n) = (r) * ((s) * (s)))))
  4. 0004specialize squarefree_decomposition_exists (n)
  5. 0005apply squarefree_decomposition_exists
  6. 0006exact hn
  7. 0007cases hexists
  8. 0008cases hexists_witness
  9. 0009exists x
  10. 0010exists x1
  11. 0011split
  12. 0012exact hexists_witness_witness
  13. 0013intro u
  14. 0014intro v
  15. 0015intro hother
  16. 0016cases hexists_witness_witness
  17. 0017cases hother
  18. 0018specialize squarefree_decomposition_functional (n)
  19. 0019specialize squarefree_decomposition_functional (x)
  20. 0020specialize squarefree_decomposition_functional (x1)
  21. 0021specialize squarefree_decomposition_functional (u)
  22. 0022specialize squarefree_decomposition_functional (v)
  23. 0023apply squarefree_decomposition_functional
  24. 0024exact hn
  25. 0025exact hexists_witness_witness_left
  26. 0026exact hother_left
  27. 0027exact hexists_witness_witness_right
  28. 0028exact hother_right