TS002R · theorem body

represented_factor_product_times_square_is_two_square

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

A represented-factor beta product remains constructively representable after multiplication by any explicitly witnessed 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

∀ b. ∀ c. ∀ l. ∀ m. ∀ z. ∀ n. Product(b,c,l,m) → (∀ x. ∀ y. Lt(x,l) → BetaAt(b,c,x,y) → ∃ k. ∃ i. y = k · k + i · i) → n = m · (z · z) → ∃ x. ∃ y. 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

PD0002 Lt PD0013 BetaAt PD0014 Product

In local proof propositions

none
Exact expanded first-order statement
forall b c l m z n. (exists ff_u_ftsf_square_product ff_v_ftsf_square_product. ((((exists ff_h_ftsf_square_product_start. ff_h_ftsf_square_product_start + S (1) = S ((S (0)) * ff_v_ftsf_square_product)) /\ exists ff_q_ftsf_square_product_start. ff_u_ftsf_square_product = ff_q_ftsf_square_product_start * S ((S (0)) * ff_v_ftsf_square_product) + (1))) /\ ((((exists ff_h_ftsf_square_product_terminal. ff_h_ftsf_square_product_terminal + S (m) = S ((S (l)) * ff_v_ftsf_square_product)) /\ exists ff_q_ftsf_square_product_terminal. ff_u_ftsf_square_product = ff_q_ftsf_square_product_terminal * S ((S (l)) * ff_v_ftsf_square_product) + (m))) /\ forall ff_i_ftsf_square_product. (exists ff_lt_ftsf_square_product_bound. ff_lt_ftsf_square_product_bound + S ff_i_ftsf_square_product = l) -> exists ff_p_ftsf_square_product ff_r_ftsf_square_product ff_s_ftsf_square_product. ((((exists ff_h_ftsf_square_product_factor. ff_h_ftsf_square_product_factor + S (ff_p_ftsf_square_product) = S ((S (ff_i_ftsf_square_product)) * c)) /\ exists ff_q_ftsf_square_product_factor. b = ff_q_ftsf_square_product_factor * S ((S (ff_i_ftsf_square_product)) * c) + (ff_p_ftsf_square_product))) /\ ((((exists ff_h_ftsf_square_product_partial. ff_h_ftsf_square_product_partial + S (ff_r_ftsf_square_product) = S ((S (ff_i_ftsf_square_product)) * ff_v_ftsf_square_product)) /\ exists ff_q_ftsf_square_product_partial. ff_u_ftsf_square_product = ff_q_ftsf_square_product_partial * S ((S (ff_i_ftsf_square_product)) * ff_v_ftsf_square_product) + (ff_r_ftsf_square_product))) /\ ((((exists ff_h_ftsf_square_product_successor. ff_h_ftsf_square_product_successor + S (ff_s_ftsf_square_product) = S ((S (S ff_i_ftsf_square_product)) * ff_v_ftsf_square_product)) /\ exists ff_q_ftsf_square_product_successor. ff_u_ftsf_square_product = ff_q_ftsf_square_product_successor * S ((S (S ff_i_ftsf_square_product)) * ff_v_ftsf_square_product) + (ff_s_ftsf_square_product))) /\ ff_s_ftsf_square_product = ff_r_ftsf_square_product * ff_p_ftsf_square_product)))))) -> (forall ftsf_index_square_prefix ftsf_factor_square_prefix. (exists ftsf_gap_square_prefix_bound. ftsf_gap_square_prefix_bound + S ftsf_index_square_prefix = (l)) -> (((exists ff_h_ftsf_square_prefix_entry. ff_h_ftsf_square_prefix_entry + S (ftsf_factor_square_prefix) = S ((S (ftsf_index_square_prefix)) * c)) /\ exists ff_q_ftsf_square_prefix_entry. b = ff_q_ftsf_square_prefix_entry * S ((S (ftsf_index_square_prefix)) * c) + (ftsf_factor_square_prefix))) -> (exists ftsf_first_square_prefix_representation ftsf_second_square_prefix_representation. (ftsf_factor_square_prefix) = ftsf_first_square_prefix_representation * ftsf_first_square_prefix_representation + ftsf_second_square_prefix_representation * ftsf_second_square_prefix_representation)) -> n = m * (z * z) -> (exists ftsf_first_square_result ftsf_second_square_result. (n) = ftsf_first_square_result * ftsf_first_square_result + ftsf_second_square_result * ftsf_second_square_result)

Proof neighborhood

Direct theorem prerequisites

TS002M beta_two_square_represented_factor_product TS001Z every_natural_square_is_sum_of_two_squares TS001X two_square_representation_multiplicatively_closed

Direct theorem dependents

TS002U beta_two_square_prefix_append_equal_pair

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

26 script commands · 3 reading checkpoints · 2 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 (3)
beta two square represented factor productevery natural square is sum of two squarestwo square representation multiplicatively closed
01Fix variables and assumptionsL1–9

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro m
  5. L5
    intro z
  6. L6
    intro n
  7. L7
    intro hproduct
  8. L8
    intro hprefix
  9. L9
    intro hdecomposition
02Establish hbaseL10–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta two square represented factor product.

  1. L10
    have hbase : exists ftsf_first_square_local_base ftsf_second_square_local_base. (m) = ftsf_first_square_local_base * ftsf_first_square_local_base + ftsf_second_square_local_base * ftsf_second_square_local_base
  2. L11
    specialize beta_two_square_represented_factor_product b
  3. L12
    specialize beta_two_square_represented_factor_product c
  4. L13
    specialize beta_two_square_represented_factor_product l
  5. L14
    specialize beta_two_square_represented_factor_product m
  6. L15
    apply beta_two_square_represented_factor_product
  7. L16
    exact hproduct
  8. L17
    exact hprefix
03Establish hsquareL18–26

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

  1. L18
    have hsquare : exists ftsf_first_square_local_factor ftsf_second_square_local_factor. (z * z) = ftsf_first_square_local_factor * ftsf_first_square_local_factor + ftsf_second_square_local_factor * ftsf_second_square_local_factor
  2. L19
    specialize every_natural_square_is_sum_of_two_squares z
  3. L20
    exact every_natural_square_is_sum_of_two_squares
  4. L21
    rewrite hdecomposition
  5. L22
    specialize two_square_representation_multiplicatively_closed m
  6. L23
    specialize two_square_representation_multiplicatively_closed (z * z)
  7. L24
    apply two_square_representation_multiplicatively_closed
  8. L25
    exact hbase
  9. L26
    exact hsquare

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro m
  5. 0005intro z
  6. 0006intro n
  7. 0007intro hproduct
  8. 0008intro hprefix
  9. 0009intro hdecomposition
  10. 0010have hbase : exists ftsf_first_square_local_base ftsf_second_square_local_base. (m) = ftsf_first_square_local_base * ftsf_first_square_local_base + ftsf_second_square_local_base * ftsf_second_square_local_base
  11. 0011specialize beta_two_square_represented_factor_product b
  12. 0012specialize beta_two_square_represented_factor_product c
  13. 0013specialize beta_two_square_represented_factor_product l
  14. 0014specialize beta_two_square_represented_factor_product m
  15. 0015apply beta_two_square_represented_factor_product
  16. 0016exact hproduct
  17. 0017exact hprefix
  18. 0018have hsquare : exists ftsf_first_square_local_factor ftsf_second_square_local_factor. (z * z) = ftsf_first_square_local_factor * ftsf_first_square_local_factor + ftsf_second_square_local_factor * ftsf_second_square_local_factor
  19. 0019specialize every_natural_square_is_sum_of_two_squares z
  20. 0020exact every_natural_square_is_sum_of_two_squares
  21. 0021rewrite hdecomposition
  22. 0022specialize two_square_representation_multiplicatively_closed m
  23. 0023specialize two_square_representation_multiplicatively_closed (z * z)
  24. 0024apply two_square_representation_multiplicatively_closed
  25. 0025exact hbase
  26. 0026exact hsquare