TF000A

three_mod_four_good_prime_exclusive

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

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

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.

The exact G025 progression-prime milestone is fully proved in unchanged constructive arithmetic; Mod4Three deliberately reuses its existing Quadratic Reciprocity definition PD0012. The much stronger full Dirichlet progression-prime milestone G030 remains open.

Exact theorem in conservative defined notation

∀ p. Prime(p) → p = 2 ∨ (∃ x. p = 4 · x + 1) → ¬Mod4Three(p)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Prime(p) · 1Mod4Three(n) · 1

Actual proof prerequisites

prime_two_or_one_mod_four_is_sum_of_two_squares · checked external prerequisitethree_mod_four_progression_not_two_square
Original expanded first-order 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

Complete unchanged native tactic proof

All 11 lines are the exact independently kernel-checked original script.

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.

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 (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 defined 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