TS002N · theorem body

beta_witnessed_two_square_factor_product

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

A represented decoded factor supplied existentially at every bounded index suffices for an explicit representation of the whole product.

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. ∀ n. Product(b,c,l,n) → (∀ x. Lt(x,l) → ∃ y. ∃ z. ∃ m. BetaAt(b,c,x,y) ∧ y = z · z + m · m) → ∃ 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 n. (exists ff_u_ftsf_witness_product ff_v_ftsf_witness_product. ((((exists ff_h_ftsf_witness_product_start. ff_h_ftsf_witness_product_start + S (1) = S ((S (0)) * ff_v_ftsf_witness_product)) /\ exists ff_q_ftsf_witness_product_start. ff_u_ftsf_witness_product = ff_q_ftsf_witness_product_start * S ((S (0)) * ff_v_ftsf_witness_product) + (1))) /\ ((((exists ff_h_ftsf_witness_product_terminal. ff_h_ftsf_witness_product_terminal + S (n) = S ((S (l)) * ff_v_ftsf_witness_product)) /\ exists ff_q_ftsf_witness_product_terminal. ff_u_ftsf_witness_product = ff_q_ftsf_witness_product_terminal * S ((S (l)) * ff_v_ftsf_witness_product) + (n))) /\ forall ff_i_ftsf_witness_product. (exists ff_lt_ftsf_witness_product_bound. ff_lt_ftsf_witness_product_bound + S ff_i_ftsf_witness_product = l) -> exists ff_p_ftsf_witness_product ff_r_ftsf_witness_product ff_s_ftsf_witness_product. ((((exists ff_h_ftsf_witness_product_factor. ff_h_ftsf_witness_product_factor + S (ff_p_ftsf_witness_product) = S ((S (ff_i_ftsf_witness_product)) * c)) /\ exists ff_q_ftsf_witness_product_factor. b = ff_q_ftsf_witness_product_factor * S ((S (ff_i_ftsf_witness_product)) * c) + (ff_p_ftsf_witness_product))) /\ ((((exists ff_h_ftsf_witness_product_partial. ff_h_ftsf_witness_product_partial + S (ff_r_ftsf_witness_product) = S ((S (ff_i_ftsf_witness_product)) * ff_v_ftsf_witness_product)) /\ exists ff_q_ftsf_witness_product_partial. ff_u_ftsf_witness_product = ff_q_ftsf_witness_product_partial * S ((S (ff_i_ftsf_witness_product)) * ff_v_ftsf_witness_product) + (ff_r_ftsf_witness_product))) /\ ((((exists ff_h_ftsf_witness_product_successor. ff_h_ftsf_witness_product_successor + S (ff_s_ftsf_witness_product) = S ((S (S ff_i_ftsf_witness_product)) * ff_v_ftsf_witness_product)) /\ exists ff_q_ftsf_witness_product_successor. ff_u_ftsf_witness_product = ff_q_ftsf_witness_product_successor * S ((S (S ff_i_ftsf_witness_product)) * ff_v_ftsf_witness_product) + (ff_s_ftsf_witness_product))) /\ ff_s_ftsf_witness_product = ff_r_ftsf_witness_product * ff_p_ftsf_witness_product)))))) -> (forall ftsf_index_witness_product_source. (exists ftsf_gap_witness_product_source_bound. ftsf_gap_witness_product_source_bound + S ftsf_index_witness_product_source = (l)) -> exists ftsf_factor_witness_product_source ftsf_coordinate_first_witness_product_source ftsf_coordinate_second_witness_product_source. ((((exists ff_h_ftsf_witness_product_source_entry. ff_h_ftsf_witness_product_source_entry + S (ftsf_factor_witness_product_source) = S ((S (ftsf_index_witness_product_source)) * c)) /\ exists ff_q_ftsf_witness_product_source_entry. b = ff_q_ftsf_witness_product_source_entry * S ((S (ftsf_index_witness_product_source)) * c) + (ftsf_factor_witness_product_source))) /\ ftsf_factor_witness_product_source = ftsf_coordinate_first_witness_product_source * ftsf_coordinate_first_witness_product_source + ftsf_coordinate_second_witness_product_source * ftsf_coordinate_second_witness_product_source)) -> (exists ftsf_first_witness_product_result ftsf_second_witness_product_result. (n) = ftsf_first_witness_product_result * ftsf_first_witness_product_result + ftsf_second_witness_product_result * ftsf_second_witness_product_result)

Proof neighborhood

Direct theorem prerequisites

TS002K beta_witnessed_two_square_prefix_implies_pointwise TS002M beta_two_square_represented_factor_product

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

17 script commands · 3 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 (2)
beta witnessed two square prefix implies pointwisebeta two square represented factor product
01Fix variables and assumptionsL1–6

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 n
  5. L5
    intro hproduct
  6. L6
    intro hwitnessed
02Use earlier factsL7–16

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

  1. L7
    specialize beta_two_square_represented_factor_product b
  2. L8
    specialize beta_two_square_represented_factor_product c
  3. L9
    specialize beta_two_square_represented_factor_product l
  4. L10
    specialize beta_two_square_represented_factor_product n
  5. L11
    apply beta_two_square_represented_factor_product
  6. L12
    exact hproduct
  7. L13
    specialize beta_witnessed_two_square_prefix_implies_pointwise b
  8. L14
    specialize beta_witnessed_two_square_prefix_implies_pointwise c
  9. L15
    specialize beta_witnessed_two_square_prefix_implies_pointwise l
  10. L16
    apply beta_witnessed_two_square_prefix_implies_pointwise
03Use earlier factsL17–17

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

  1. L17
    exact hwitnessed

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro n
  5. 0005intro hproduct
  6. 0006intro hwitnessed
  7. 0007specialize beta_two_square_represented_factor_product b
  8. 0008specialize beta_two_square_represented_factor_product c
  9. 0009specialize beta_two_square_represented_factor_product l
  10. 0010specialize beta_two_square_represented_factor_product n
  11. 0011apply beta_two_square_represented_factor_product
  12. 0012exact hproduct
  13. 0013specialize beta_witnessed_two_square_prefix_implies_pointwise b
  14. 0014specialize beta_witnessed_two_square_prefix_implies_pointwise c
  15. 0015specialize beta_witnessed_two_square_prefix_implies_pointwise l
  16. 0016apply beta_witnessed_two_square_prefix_implies_pointwise
  17. 0017exact hwitnessed