TS002H · theorem body

two_square_product_is_two_square

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

The complete constructive Brahmagupta--Fibonacci identity supplies actual natural coordinates for every product of two-square norms.

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. exists x y. (a * a + b * b) * (c * c + d * d) = x * x + y * y

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. exists x y. (a * a + b * b) * (c * c + d * d) = x * x + y * y

Proof neighborhood

Direct theorem prerequisites

TS002G two_square_product_explicit_witness

Direct theorem dependents

TS002I two_square_representations_closed_under_multiplication

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

13 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
two square product explicit witness
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
02Use earlier factsL5–8

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

  1. L5
    specialize two_square_product_explicit_witness a
  2. L6
    specialize two_square_product_explicit_witness b
  3. L7
    specialize two_square_product_explicit_witness c
  4. L8
    specialize two_square_product_explicit_witness d
03Separate the logical casesL9–10

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases two_square_product_explicit_witness
  2. L10
    cases two_square_product_explicit_witness_witness
04Construct an explicit witnessL11–12

Supply the displayed value, then prove that it has the required property.

  1. L11
    exists a * c + b * d
  2. L12
    exists x
05Use earlier factsL13–13

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

  1. L13
    exact two_square_product_explicit_witness_witness_right

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005specialize two_square_product_explicit_witness a
  6. 0006specialize two_square_product_explicit_witness b
  7. 0007specialize two_square_product_explicit_witness c
  8. 0008specialize two_square_product_explicit_witness d
  9. 0009cases two_square_product_explicit_witness
  10. 0010cases two_square_product_explicit_witness_witness
  11. 0011exists a * c + b * d
  12. 0012exists x
  13. 0013exact two_square_product_explicit_witness_witness_right