FS0019

four_square_descent_product_reassociate

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

The product of two k-divisible norms exposes its exact common square factor k².

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 k r. (p * k) * (k * r) = (k * k) * (p * r)

Constructive proof overview

Generated structural guide

The product of two k-divisible norms exposes its exact common square factor k².

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

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

mul_comm Stable theorem; checked-use authorized mul_shuffle_four Alpha theorem; checked-use authorized

Direct dependents

FS001C four_square_descent_scaled_norm_quotient

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 · 5 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 k
  3. L3
    intro r
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 (k * p) * (k * r)
  2. L5
    congr
03Use earlier factsL6–6

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

  1. L6
    apply mul_comm
04Calculate and transport equalitiesL7–7

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

  1. L7
    refl
05Use earlier factsL8–8

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

  1. L8
    apply mul_shuffle_four

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro r
  4. 0004trans (k * p) * (k * r)
  5. 0005congr
  6. 0006apply mul_comm
  7. 0007refl
  8. 0008apply mul_shuffle_four