TF0012

infinitely_many_primes_three_mod_four

For every natural bound, construct an actual strictly larger prime with an explicit residue witness p=4k+3.

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

∀ B. ∃ p. Prime(p) ∧ (Lt(B,p) ∧ Mod4Three(p))

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

Definition DAG

Lt(a,b) · 1Prime(p) · 1Mod4Three(n) · 1

Actual proof prerequisites

bounded_common_multiple_exists · checked external prerequisitenonzero_is_succ · checked external prerequisiteeuclid_three_progression_prime_existseuclid_three_prime_divisor_exceeds_bound
Original expanded first-order statement 
forall B. exists 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)) /\ ((exists gap. gap + S B = p) /\ (exists ff_residue_ptmf_prime. (p) = 4 * ff_residue_ptmf_prime + 3)))

Complete unchanged native tactic proof

All 27 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

27 script commands · 12 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.

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 (2)
euclid three progression prime existseuclid three prime divisor exceeds bound
01Fix variables and assumptionsL1–1

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

  1. L1
    intro B
02Use earlier factsL2–2

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

  1. L2
    specialize bounded_common_multiple_exists B
03Separate the logical casesL3–4

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L3
    cases bounded_common_multiple_exists
  2. L4
    cases bounded_common_multiple_exists_witness
04Establish hpredecessorL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L5
    have hpredecessor : exists d. x = S d
  2. L6
    specialize nonzero_is_succ x
  3. L7
    apply nonzero_is_succ
  4. L8
    exact bounded_common_multiple_exists_witness_left
05Separate the logical casesL9–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases hpredecessor
06Use earlier factsL10–10

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

  1. L10
    specialize euclid_three_progression_prime_exists x1
07Separate the logical casesL11–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases euclid_three_progression_prime_exists
  2. L12
    cases euclid_three_progression_prime_exists_witness
  3. L13
    cases euclid_three_progression_prime_exists_witness_right
08Construct an explicit witnessL14–14

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

  1. L14
    exists x2
09Separate the logical casesL15–15

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L15
    split
10Use earlier factsL16–16

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

  1. L16
    exact euclid_three_progression_prime_exists_witness_left
11Separate the logical casesL17–17

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L17
    split
12Use earlier factsL18–27

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

  1. L18
    specialize euclid_three_prime_divisor_exceeds_bound B
  2. L19
    specialize euclid_three_prime_divisor_exceeds_bound x
  3. L20
    specialize euclid_three_prime_divisor_exceeds_bound x1
  4. L21
    specialize euclid_three_prime_divisor_exceeds_bound x2
  5. L22
    apply euclid_three_prime_divisor_exceeds_bound
  6. L23
    exact bounded_common_multiple_exists_witness_right
  7. L24
    exact hpredecessor_witness
  8. L25
    exact euclid_three_progression_prime_exists_witness_left
  9. L26
    exact euclid_three_progression_prime_exists_witness_right_right
  10. L27
    exact euclid_three_progression_prime_exists_witness_right_left

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro B
  2. 0002specialize bounded_common_multiple_exists B
  3. 0003cases bounded_common_multiple_exists
  4. 0004cases bounded_common_multiple_exists_witness
  5. 0005have hpredecessor : exists d. x = S d
  6. 0006specialize nonzero_is_succ x
  7. 0007apply nonzero_is_succ
  8. 0008exact bounded_common_multiple_exists_witness_left
  9. 0009cases hpredecessor
  10. 0010specialize euclid_three_progression_prime_exists x1
  11. 0011cases euclid_three_progression_prime_exists
  12. 0012cases euclid_three_progression_prime_exists_witness
  13. 0013cases euclid_three_progression_prime_exists_witness_right
  14. 0014exists x2
  15. 0015split
  16. 0016exact euclid_three_progression_prime_exists_witness_left
  17. 0017split
  18. 0018specialize euclid_three_prime_divisor_exceeds_bound B
  19. 0019specialize euclid_three_prime_divisor_exceeds_bound x
  20. 0020specialize euclid_three_prime_divisor_exceeds_bound x1
  21. 0021specialize euclid_three_prime_divisor_exceeds_bound x2
  22. 0022apply euclid_three_prime_divisor_exceeds_bound
  23. 0023exact bounded_common_multiple_exists_witness_right
  24. 0024exact hpredecessor_witness
  25. 0025exact euclid_three_progression_prime_exists_witness_left
  26. 0026exact euclid_three_progression_prime_exists_witness_right_right
  27. 0027exact euclid_three_progression_prime_exists_witness_right_left