TS003N

three_mod_four_prime_two_square_norm_extracts_squared_factor

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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)))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 21 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

TS0027 three_mod_four_prime_divides_two_square_norm_divides_both TS003K two_square_common_divisor_extracts_squared_factor

Direct dependents

TS003O three_mod_four_prime_nonzero_two_square_norm_extracts_nonzero_quotient

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 : (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)
  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 exact 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 : (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