SK0010

squarefree_decomposition_exists_unique

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.

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. ¬n = 0 → ∃ x. ∃ y. NaturalSquarefreeDecomposition(n,x,y) ∧ (∀ z. ∀ m. NaturalSquarefreeDecomposition(n,z,m) → z = x ∧ m = y)

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

Definition DAG

NaturalSquarefreeDecomposition(n,r,s) · 3

Actual proof prerequisites

squarefree_decomposition_existssquarefree_decomposition_functional
Original expanded first-order 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

Complete tactic proof in conservative notation

All 28 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

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.

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 decomposition existssquarefree decomposition functional
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(n,r,s)Original native command in the exact edition
  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 defined command ledger · 28 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003have hexists : ∃ r. ∃ s. NaturalSquarefreeDecomposition(n,r,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