TF000A

three_mod_four_good_prime_exclusive

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

A prime equal to two or one modulo four cannot also be three modulo four.

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. ((~(p = 1) /\ forall frm_prime_left_ptmf_prime frm_prime_right_ptmf_prime. p = frm_prime_left_ptmf_prime * frm_prime_right_ptmf_prime -> frm_prime_left_ptmf_prime = 1 \/ frm_prime_right_ptmf_prime = 1)) -> (p = 2 \/ exists k. p = 4 * k + 1) -> (exists ff_residue_ptmf_prime. (p) = 4 * ff_residue_ptmf_prime + 3) -> false

Constructive proof overview

Generated structural guide

A prime equal to two or one modulo four cannot also be three modulo four.

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v23 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_two_or_one_mod_four_is_sum_of_two_squares Alpha theorem; checked-use authorized TF0009 three_mod_four_progression_not_two_square

Direct dependents

TF000B three_mod_four_prime_divisor_decidable

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

11 script commands · 2 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.

Named ingredients (1)
three mod four progression not two square
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro hprime
  3. L3
    intro hgood
  4. L4
    intro hthree
02Use earlier factsL5–11

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

  1. L5
    specialize three_mod_four_progression_not_two_square p
  2. L6
    apply three_mod_four_progression_not_two_square
  3. L7
    exact hthree
  4. L8
    specialize prime_two_or_one_mod_four_is_sum_of_two_squares p
  5. L9
    apply prime_two_or_one_mod_four_is_sum_of_two_squares
  6. L10
    exact hprime
  7. L11
    exact hgood

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003intro hgood
  4. 0004intro hthree
  5. 0005specialize three_mod_four_progression_not_two_square p
  6. 0006apply three_mod_four_progression_not_two_square
  7. 0007exact hthree
  8. 0008specialize prime_two_or_one_mod_four_is_sum_of_two_squares p
  9. 0009apply prime_two_or_one_mod_four_is_sum_of_two_squares
  10. 0010exact hprime
  11. 0011exact hgood