SK000C

squarefree_decomposition_exists

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

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

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_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)))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 8 exact native proof lines.

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

Proof neighborhood

Direct dependencies

SK000B squarefree_decomposition_bounded_exists le_refl Stable theorem; checked-use authorized

Direct dependents

SK0010 squarefree_decomposition_exists_unique

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

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.

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 exact 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