TF000F

euclid_three_progression_prime_exists

Every actual subtraction-free Euclid number 4d+3 has a witnessed prime divisor congruent to 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

∀ d. ∃ p. PrimeThreeModFourDivisor(4 · d + 3,p)

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

Definition DAG

PrimeThreeModFourDivisor(n,p) · 1

Actual proof prerequisites

three_mod_four_prime_divisor_exists
Original expanded first-order statement 
forall d. exists p. ((((~(p = 1) /\ forall frm_prime_left_ptmf_euclid_prime frm_prime_right_ptmf_euclid_prime. p = frm_prime_left_ptmf_euclid_prime * frm_prime_right_ptmf_euclid_prime -> frm_prime_left_ptmf_euclid_prime = 1 \/ frm_prime_right_ptmf_euclid_prime = 1)) /\ ((exists ff_residue_ptmf_euclid_residue. (p) = 4 * ff_residue_ptmf_euclid_residue + 3) /\ (exists ff_quotient_ptmf_euclid_divides. (4 * d + 3) = (p) * ff_quotient_ptmf_euclid_divides))))

Complete unchanged native tactic proof

All 5 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

5 script commands · 4 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 prime divisor exists
01Fix variables and assumptionsL1–1

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

  1. L1
    intro d
02Use earlier factsL2–3

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

  1. L2
    specialize three_mod_four_prime_divisor_exists (4 * d + 3)
  2. L3
    apply three_mod_four_prime_divisor_exists
03Construct an explicit witnessL4–4

Supply the displayed value, then prove that it has the required property.

  1. L4
    exists d
04Calculate and transport equalitiesL5–5

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L5
    refl

Library-wide reading audit

Original defined command ledger · 5 lines
  1. 0001intro d
  2. 0002specialize three_mod_four_prime_divisor_exists (4 * d + 3)
  3. 0003apply three_mod_four_prime_divisor_exists
  4. 0004exists d
  5. 0005refl