TF0011

euclid_three_prime_divisor_exceeds_bound

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

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

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 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 29 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_or_lt Stable theorem; checked-use authorized bounded_common_multiple_contains_bounded_prime Alpha theorem; checked-use authorized TF0010 euclid_three_common_multiple_exclusion

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

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.

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 exact 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