TS001U

two_square_product_norm_blocks

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

The four squared products regroup into diagonal and off-diagonal norm blocks.

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 a b c d. (a * a + b * b) * (c * c + d * d) = ((a * c) * (a * c) + (b * d) * (b * d)) + ((a * d) * (a * d) + (b * c) * (b * c))

Constructive proof overview

Generated structural guide

The four squared products regroup into diagonal and off-diagonal norm blocks.

The unchanged tactic script uses 3 declared prerequisites and contains 11 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

TS001T two_square_product_expands add_comm Stable theorem; checked-use authorized add_shuffle_middle Alpha theorem; checked-use authorized

Direct dependents

TS001W brahmagupta_fibonacci_two_square_identity

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

11 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.

Named ingredients (1)
two square product expands
01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
02Calculate and transport equalitiesL5–5

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

  1. L5
    trans ((a * c) * (a * c) + (a * d) * (a * d)) + ((b * c) * (b * c) + (b * d) * (b * d))
03Use earlier factsL6–6

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

  1. L6
    apply two_square_product_expands
04Calculate and transport equalitiesL7–9

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

  1. L7
    trans ((a * c) * (a * c) + (a * d) * (a * d)) + ((b * d) * (b * d) + (b * c) * (b * c))
  2. L8
    congr
  3. L9
    refl
05Use earlier factsL10–11

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

  1. L10
    apply add_comm
  2. L11
    apply add_shuffle_middle

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005trans ((a * c) * (a * c) + (a * d) * (a * d)) + ((b * c) * (b * c) + (b * d) * (b * d))
  6. 0006apply two_square_product_expands
  7. 0007trans ((a * c) * (a * c) + (a * d) * (a * d)) + ((b * d) * (b * d) + (b * c) * (b * c))
  8. 0008congr
  9. 0009refl
  10. 0010apply add_comm
  11. 0011apply add_shuffle_middle