TS003N · theorem body

three_mod_four_prime_two_square_norm_extracts_squared_factor

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

A three-modulo-four prime dividing a two-square norm extracts as an exact prime square while preserving a represented quotient.

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

∀ p. ∀ a. ∀ b. Prime(p) → Mod4Three(p) → Dvd(p,a · a + b · b) → ∃ x. ∃ y. a = p · x ∧ (b = p · y ∧ a · a + b · b = p · p · (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

PD0003 Dvd PD0004 Prime PD0012 Mod4Three

In local proof propositions

PD0003 Dvd
Exact expanded first-order statement
forall p a b. ((~(p = 1) /\ forall frm_prime_left_ftsv_prime frm_prime_right_ftsv_prime. p = frm_prime_left_ftsv_prime * frm_prime_right_ftsv_prime -> frm_prime_left_ftsv_prime = 1 \/ frm_prime_right_ftsv_prime = 1)) -> (exists ftsc_four_three_ftsv_prime. (p) = 4 * ftsc_four_three_ftsv_prime + 3) -> (exists ftcn_factor_ftsv_norm_divides. (a * a + b * b) = (p) * ftcn_factor_ftsv_norm_divides) -> (exists ftsv_first_extracted ftsv_second_extracted. ((a = p * ftsv_first_extracted) /\ ((b = p * ftsv_second_extracted) /\ (a * a + b * b = (p * p) * (ftsv_first_extracted * ftsv_first_extracted + ftsv_second_extracted * ftsv_second_extracted)))))

Proof neighborhood

Direct theorem prerequisites

TS0027 three_mod_four_prime_divides_two_square_norm_divides_both TS003K two_square_common_divisor_extracts_squared_factor

Direct theorem dependents

TS003O three_mod_four_prime_nonzero_two_square_norm_extracts_nonzero_quotient

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

21 script commands · 4 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)
three mod four prime divides two square norm divides bothtwo square common divisor extracts squared factor
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hprime
  5. L5
    intro hthree
  6. L6
    intro hnorm
02Establish hbothL7–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply three mod four prime divides two square norm divides both.

  1. L7
    have hboth : Dvd(p,a) ∧ Dvd(p,b)Definitions: Dvd(p,a)Dvd(p,b)Original native command in the exact edition
  2. L8
    specialize three_mod_four_prime_divides_two_square_norm_divides_both p
  3. L9
    specialize three_mod_four_prime_divides_two_square_norm_divides_both a
  4. L10
    specialize three_mod_four_prime_divides_two_square_norm_divides_both b
  5. L11
    apply three_mod_four_prime_divides_two_square_norm_divides_both
  6. L12
    exact hprime
  7. L13
    exact hthree
  8. L14
    exact hnorm
03Separate the logical casesL15–15

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

  1. L15
    cases hboth
04Use earlier factsL16–21

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

  1. L16
    specialize two_square_common_divisor_extracts_squared_factor p
  2. L17
    specialize two_square_common_divisor_extracts_squared_factor a
  3. L18
    specialize two_square_common_divisor_extracts_squared_factor b
  4. L19
    apply two_square_common_divisor_extracts_squared_factor
  5. L20
    exact hboth_left
  6. L21
    exact hboth_right

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hprime
  5. 0005intro hthree
  6. 0006intro hnorm
  7. 0007have hboth : Dvd(p,a) ∧ Dvd(p,b)
    Exact native replay linehave hboth : (exists ftcn_factor_ftsv_first_divides. (a) = (p) * ftcn_factor_ftsv_first_divides) /\ (exists ftcn_factor_ftsv_second_divides. (b) = (p) * ftcn_factor_ftsv_second_divides)
  8. 0008specialize three_mod_four_prime_divides_two_square_norm_divides_both p
  9. 0009specialize three_mod_four_prime_divides_two_square_norm_divides_both a
  10. 0010specialize three_mod_four_prime_divides_two_square_norm_divides_both b
  11. 0011apply three_mod_four_prime_divides_two_square_norm_divides_both
  12. 0012exact hprime
  13. 0013exact hthree
  14. 0014exact hnorm
  15. 0015cases hboth
  16. 0016specialize two_square_common_divisor_extracts_squared_factor p
  17. 0017specialize two_square_common_divisor_extracts_squared_factor a
  18. 0018specialize two_square_common_divisor_extracts_squared_factor b
  19. 0019apply two_square_common_divisor_extracts_squared_factor
  20. 0020exact hboth_left
  21. 0021exact hboth_right