TS003M · theorem body

two_square_representation_preserved_by_square_factor

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

Every represented natural remains represented after multiplication by an arbitrary natural square.

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 n z. (exists ftsc_first_ftsv_scale_source ftsc_second_ftsv_scale_source. (n) = ftsc_first_ftsv_scale_source * ftsc_first_ftsv_scale_source + ftsc_second_ftsv_scale_source * ftsc_second_ftsv_scale_source) -> (exists ftsc_first_ftsv_scale_result ftsc_second_ftsv_scale_result. (n * (z * z)) = ftsc_first_ftsv_scale_result * ftsc_first_ftsv_scale_result + ftsc_second_ftsv_scale_result * ftsc_second_ftsv_scale_result)

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 n z. (exists ftsc_first_ftsv_scale_source ftsc_second_ftsv_scale_source. (n) = ftsc_first_ftsv_scale_source * ftsc_first_ftsv_scale_source + ftsc_second_ftsv_scale_source * ftsc_second_ftsv_scale_source) -> (exists ftsc_first_ftsv_scale_result ftsc_second_ftsv_scale_result. (n * (z * z)) = ftsc_first_ftsv_scale_result * ftsc_first_ftsv_scale_result + ftsc_second_ftsv_scale_result * ftsc_second_ftsv_scale_result)

Proof neighborhood

Direct theorem prerequisites

TS001Z every_natural_square_is_sum_of_two_squares TS001X two_square_representation_multiplicatively_closed

Direct theorem dependents

TS003C all_bad_prime_even_two_square_sufficiency_bounded

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

11 script commands · 2 reading checkpoints · 1 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 (2)
every natural square is sum of two squarestwo square representation multiplicatively closed
01Fix variables and assumptionsL1–3

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

  1. L1
    intro n
  2. L2
    intro z
  3. L3
    intro hrepresented
02Establish hsquareL4–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply two square representation multiplicatively closed.

  1. L4
    have hsquare : exists ftsc_first_ftsv_scale_witness ftsc_second_ftsv_scale_witness. (z * z) = ftsc_first_ftsv_scale_witness * ftsc_first_ftsv_scale_witness + ftsc_second_ftsv_scale_witness * ftsc_second_ftsv_scale_witness
  2. L5
    specialize every_natural_square_is_sum_of_two_squares z
  3. L6
    exact every_natural_square_is_sum_of_two_squares
  4. L7
    specialize two_square_representation_multiplicatively_closed n
  5. L8
    specialize two_square_representation_multiplicatively_closed (z * z)
  6. L9
    apply two_square_representation_multiplicatively_closed
  7. L10
    exact hrepresented
  8. L11
    exact hsquare

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro n
  2. 0002intro z
  3. 0003intro hrepresented
  4. 0004have hsquare : exists ftsc_first_ftsv_scale_witness ftsc_second_ftsv_scale_witness. (z * z) = ftsc_first_ftsv_scale_witness * ftsc_first_ftsv_scale_witness + ftsc_second_ftsv_scale_witness * ftsc_second_ftsv_scale_witness
  5. 0005specialize every_natural_square_is_sum_of_two_squares z
  6. 0006exact every_natural_square_is_sum_of_two_squares
  7. 0007specialize two_square_representation_multiplicatively_closed n
  8. 0008specialize two_square_representation_multiplicatively_closed (z * z)
  9. 0009apply two_square_representation_multiplicatively_closed
  10. 0010exact hrepresented
  11. 0011exact hsquare