SK0007

squarefree_square_factor_reassociate

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

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

∀ p. ∀ r. ∀ s. p · p · (r · (s · s)) = r · (p · s · (p · s))

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

Definition DAG

none

Actual proof prerequisites

mul_assoc · checked external prerequisitemul_comm · checked external prerequisitefour_square_product_square · checked external prerequisite
Original expanded first-order statement 
forall p r s. (p * p) * (r * (s * s)) = r * ((p * s) * (p * s))

Complete tactic proof in conservative notation

All 16 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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