TS002C · theorem body

two_square_product_norm_expanded

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

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.

Statement with defined notation

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))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order 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))

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed TS0029 two_square_add_left_comm mul_assoc · Stable closed mul_comm · Stable closed TS002A two_square_mul_left_comm mul_add · Stable closed add_mul · Stable closed

Direct theorem dependents

TS002E two_square_product_difference_forward TS002F two_square_product_difference_reverse

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

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 assumptionsL1–4

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 defined 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]