TS002C

two_square_product_norm_expanded

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

The product of two natural two-square norms expands into its four squared coordinate products.

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 product of two natural two-square norms expands into its four squared coordinate products.

The unchanged tactic script uses 8 declared prerequisites and contains 5 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

add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized TS0029 two_square_add_left_comm mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized TS002A two_square_mul_left_comm mul_add Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized

Direct dependents

TS002E two_square_product_difference_forward TS002F two_square_product_difference_reverse

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

5 script commands · 2 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 assumptionsL1–4

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
    simp [mul_add, add_mul, add_assoc, add_comm, two_square_add_left_comm, mul_assoc, mul_comm, two_square_mul_left_comm]

Library-wide reading audit

Original exact command ledger · 5 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005simp [mul_add, add_mul, add_assoc, add_comm, two_square_add_left_comm, mul_assoc, mul_comm, two_square_mul_left_comm]