SK000C

squarefree_decomposition_exists

Every positive natural is an actual squarefree natural times an actual natural square.

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)

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

Definition DAG

NaturalSquarefreeDecomposition(n,r,s) · 1

Actual proof prerequisites

squarefree_decomposition_bounded_existsle_refl · checked external prerequisite
Original expanded first-order statement 
forall n. ~(n = 0) -> exists r s. (((((~((r) = 0)) /\ (forall sfd_prime_unbounded_decompositionkernel. (~((sfd_prime_unbounded_decompositionkernel) = 1) /\ forall pvs_left_unbounded_decompositionkerneldomain pvs_right_unbounded_decompositionkerneldomain. (sfd_prime_unbounded_decompositionkernel) = pvs_left_unbounded_decompositionkerneldomain * pvs_right_unbounded_decompositionkerneldomain -> pvs_left_unbounded_decompositionkerneldomain = 1 \/ pvs_right_unbounded_decompositionkerneldomain = 1) -> (exists pvs_le_gap_unbounded_decompositionkernelbound. pvs_le_gap_unbounded_decompositionkernelbound + (sfd_prime_unbounded_decompositionkernel) = (r)) -> ~(exists pvs_factor_unbounded_decompositionkernelsquare. (r) = (sfd_prime_unbounded_decompositionkernel * sfd_prime_unbounded_decompositionkernel) * pvs_factor_unbounded_decompositionkernelsquare)))) /\ ((n) = (r) * ((s) * (s)))))

Complete tactic proof in conservative notation

All 8 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

8 script commands · 2 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.

Named ingredients (1)
squarefree decomposition bounded exists
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hn
02Use earlier factsL3–8

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

  1. L3
    specialize squarefree_decomposition_bounded_exists (S n)
  2. L4
    specialize squarefree_decomposition_bounded_exists (n)
  3. L5
    apply squarefree_decomposition_bounded_exists
  4. L6
    exact hn
  5. L7
    specialize le_refl (S n)
  6. L8
    apply le_refl

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003specialize squarefree_decomposition_bounded_exists (S n)
  4. 0004specialize squarefree_decomposition_bounded_exists (n)
  5. 0005apply squarefree_decomposition_bounded_exists
  6. 0006exact hn
  7. 0007specialize le_refl (S n)
  8. 0008apply le_refl