SK0007

squarefree_square_factor_reassociate

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

Restoring a prime-square factor multiplies the actual square root and preserves the squarefree kernel.

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 p r s. (p * p) * (r * (s * s)) = r * ((p * s) * (p * s))

Constructive proof overview

Generated structural guide

Restoring a prime-square factor multiplies the actual square root and preserves the squarefree kernel.

The unchanged tactic script uses 3 declared prerequisites and contains 16 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized four_square_product_square Alpha theorem; checked-use authorized

Direct dependents

SK000B squarefree_decomposition_bounded_exists SK000F squarefree_decomposition_functional

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

16 script commands · 9 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro r
  3. L3
    intro s
02Calculate and transport equalitiesL4–5

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

  1. L4
    trans ((p * p) * r) * (s * s)
  2. L5
    symm
03Use earlier factsL6–6

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

  1. L6
    apply mul_assoc
04Calculate and transport equalitiesL7–8

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

  1. L7
    trans (r * (p * p)) * (s * s)
  2. L8
    congr
05Use earlier factsL9–9

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

  1. L9
    apply mul_comm
06Calculate and transport equalitiesL10–11

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

  1. L10
    refl
  2. L11
    trans r * ((p * p) * (s * s))
07Use earlier factsL12–12

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

  1. L12
    apply mul_assoc
08Calculate and transport equalitiesL13–15

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

  1. L13
    congr
  2. L14
    refl
  3. L15
    symm
09Use earlier factsL16–16

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

  1. L16
    apply four_square_product_square

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro p
  2. 0002intro r
  3. 0003intro s
  4. 0004trans ((p * p) * r) * (s * s)
  5. 0005symm
  6. 0006apply mul_assoc
  7. 0007trans (r * (p * p)) * (s * s)
  8. 0008congr
  9. 0009apply mul_comm
  10. 0010refl
  11. 0011trans r * ((p * p) * (s * s))
  12. 0012apply mul_assoc
  13. 0013congr
  14. 0014refl
  15. 0015symm
  16. 0016apply four_square_product_square