TF0011

euclid_three_prime_divisor_exceeds_bound

Every prime divisor of the exact Euclid number 4c−1 lies strictly above the bound encoded by its nonzero common multiple c.

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. ∀ c. ∀ d. ∀ p. (∀ x. (∃ y. S x + S y = S B) → Dvd(S x,c)) → c = S d → Prime(p) → Dvd(p,4 · d + 3) → Lt(B,p)

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

Definition DAG

Lt(a,b) · 1Dvd(d,n) · 2Prime(p) · 1

Actual proof prerequisites

le_or_lt · checked external prerequisitebounded_common_multiple_contains_bounded_prime · checked external prerequisiteeuclid_three_common_multiple_exclusion
Original expanded first-order statement 
forall B c d p. (forall ptmf_predecessor_source. (exists ptmf_common_gap_source. S ptmf_predecessor_source + S ptmf_common_gap_source = S B) -> exists ptmf_common_quotient_source. c = S ptmf_predecessor_source * ptmf_common_quotient_source) -> c = S d -> ((~(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 ff_quotient_ptmf_euclid. (4 * d + 3) = (p) * ff_quotient_ptmf_euclid) -> (exists gap. gap + S B = p)

Complete unchanged native tactic proof

All 29 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

29 script commands · 5 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 (1)
euclid three common multiple exclusion
01Fix variables and assumptionsL1–8

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

  1. L1
    intro B
  2. L2
    intro c
  3. L3
    intro d
  4. L4
    intro p
  5. L5
    intro hcommon
  6. L6
    intro hpredecessor
  7. L7
    intro hprime
  8. L8
    intro heuclid
02Use earlier factsL9–10

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

  1. L9
    specialize le_or_lt p
  2. L10
    specialize le_or_lt B
03Separate the logical casesL11–12

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

  1. L11
    cases le_or_lt
  2. L12
    exfalso
04Establish hdividesL13–22

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply bounded common multiple contains bounded prime.

  1. L13
    have hdivides : exists q. c = p * q
  2. L14
    specialize bounded_common_multiple_contains_bounded_prime B
  3. L15
    specialize bounded_common_multiple_contains_bounded_prime c
  4. L16
    specialize bounded_common_multiple_contains_bounded_prime p
  5. L17
    apply bounded_common_multiple_contains_bounded_prime
  6. L18
    exact hcommon
  7. L19
    exact hprime
  8. L20
    exact le_or_lt_left
  9. L21
    specialize euclid_three_common_multiple_exclusion c
  10. L22
    specialize euclid_three_common_multiple_exclusion d
05Use earlier factsL23–29

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

  1. L23
    specialize euclid_three_common_multiple_exclusion p
  2. L24
    apply euclid_three_common_multiple_exclusion
  3. L25
    exact hpredecessor
  4. L26
    exact hprime
  5. L27
    exact hdivides
  6. L28
    exact heuclid
  7. L29
    exact le_or_lt_right

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro B
  2. 0002intro c
  3. 0003intro d
  4. 0004intro p
  5. 0005intro hcommon
  6. 0006intro hpredecessor
  7. 0007intro hprime
  8. 0008intro heuclid
  9. 0009specialize le_or_lt p
  10. 0010specialize le_or_lt B
  11. 0011cases le_or_lt
  12. 0012exfalso
  13. 0013have hdivides : exists q. c = p * q
  14. 0014specialize bounded_common_multiple_contains_bounded_prime B
  15. 0015specialize bounded_common_multiple_contains_bounded_prime c
  16. 0016specialize bounded_common_multiple_contains_bounded_prime p
  17. 0017apply bounded_common_multiple_contains_bounded_prime
  18. 0018exact hcommon
  19. 0019exact hprime
  20. 0020exact le_or_lt_left
  21. 0021specialize euclid_three_common_multiple_exclusion c
  22. 0022specialize euclid_three_common_multiple_exclusion d
  23. 0023specialize euclid_three_common_multiple_exclusion p
  24. 0024apply euclid_three_common_multiple_exclusion
  25. 0025exact hpredecessor
  26. 0026exact hprime
  27. 0027exact hdivides
  28. 0028exact heuclid
  29. 0029exact le_or_lt_right