TS002I · theorem body

two_square_representations_closed_under_multiplication

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

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.

Statement with defined notation

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

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 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

Proof neighborhood

Direct theorem prerequisites

TS002H two_square_product_is_two_square

Direct theorem dependents

none

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

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.

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 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 defined 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