TS002I

two_square_representations_closed_under_multiplication

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

Two explicitly represented sums of two natural squares multiply to another explicitly represented sum of two squares.

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 m n. (exists a b. m = a * a + b * b) -> (exists c d. n = c * c + d * d) -> exists x y. m * n = x * x + y * y

Constructive proof overview

Generated structural guide

Two explicitly represented sums of two natural squares multiply to another explicitly represented sum of two squares.

The unchanged tactic script uses 1 declared prerequisite and contains 15 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

TS002H two_square_product_is_two_square

Direct dependents

none

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

15 script commands · 4 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 is two square
01Fix variables and assumptionsL1–4

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

  1. L1
    intro m
  2. L2
    intro n
  3. L3
    intro hfirst
  4. L4
    intro hsecond
02Separate the logical casesL5–8

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

  1. L5
    cases hfirst
  2. L6
    cases hfirst_witness
  3. L7
    cases hsecond
  4. L8
    cases hsecond_witness
03Calculate and transport equalitiesL9–10

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

  1. L9
    rewrite hfirst_witness_witness
  2. L10
    rewrite hsecond_witness_witness
04Use earlier factsL11–15

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

  1. L11
    specialize two_square_product_is_two_square x
  2. L12
    specialize two_square_product_is_two_square x1
  3. L13
    specialize two_square_product_is_two_square x2
  4. L14
    specialize two_square_product_is_two_square x3
  5. L15
    exact two_square_product_is_two_square

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro m
  2. 0002intro n
  3. 0003intro hfirst
  4. 0004intro hsecond
  5. 0005cases hfirst
  6. 0006cases hfirst_witness
  7. 0007cases hsecond
  8. 0008cases hsecond_witness
  9. 0009rewrite hfirst_witness_witness
  10. 0010rewrite hsecond_witness_witness
  11. 0011specialize two_square_product_is_two_square x
  12. 0012specialize two_square_product_is_two_square x1
  13. 0013specialize two_square_product_is_two_square x2
  14. 0014specialize two_square_product_is_two_square x3
  15. 0015exact two_square_product_is_two_square