PF0027

fermat_four_scaled_fourth_identity

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

Scaling a fourth power exposes the fourth power of its scale by two explicit product-square identities.

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 d a. ((d * a) * (d * a)) * ((d * a) * (d * a)) = ((d * d) * (d * d)) * ((a * a) * (a * a))

Constructive proof overview

Generated structural guide

Scaling a fourth power exposes the fourth power of its scale by two explicit product-square identities.

The unchanged tactic script uses 1 declared prerequisite and contains 7 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

Proof neighborhood

Direct dependencies

four_square_product_square Alpha theorem; checked-use authorized

Direct dependents

PF0028 fermat_four_scaled_equation

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

7 script commands · 3 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 assumptionsL1–2

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

  1. L1
    intro d
  2. L2
    intro a
02Calculate and transport equalitiesL3–4

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

  1. L3
    trans ((d * d) * (a * a)) * ((d * d) * (a * a))
  2. L4
    congr
03Use earlier factsL5–7

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

  1. L5
    apply four_square_product_square
  2. L6
    apply four_square_product_square
  3. L7
    apply four_square_product_square

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro d
  2. 0002intro a
  3. 0003trans ((d * d) * (a * a)) * ((d * d) * (a * a))
  4. 0004congr
  5. 0005apply four_square_product_square
  6. 0006apply four_square_product_square
  7. 0007apply four_square_product_square