TF000F

euclid_three_progression_prime_exists

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

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 5 exact native proof lines.

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

Proof neighborhood

Direct dependencies

TF000D three_mod_four_prime_divisor_exists

Direct dependents

TF0012 infinitely_many_primes_three_mod_four

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

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.

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 exact 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