FS000N · theorem body

four_square_odd_prime_bounded_modular_seed

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

Every odd prime has an actual modular square seed with a strictly smaller natural multiplier.

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.

Statement with defined notation

∀ p. ∀ h. p = 2 · h + 1 → Prime(p) → ∃ x. ∃ y. ∃ z. x · x + y · y + 1 = p · z ∧ Lt(z,p)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0004 Prime

In local proof propositions

PD0001 Le
Exact expanded first-order statement
forall p h. p = 2 * h + 1 -> ((~(p = 1) /\ forall frm_prime_left_fsbs_prime frm_prime_right_fsbs_prime. p = frm_prime_left_fsbs_prime * frm_prime_right_fsbs_prime -> frm_prime_left_fsbs_prime = 1 \/ frm_prime_right_fsbs_prime = 1)) -> (exists fsbs_a_odd_bounded fsbs_b_odd_bounded fsbs_k_odd_bounded. ((fsbs_a_odd_bounded * fsbs_a_odd_bounded + fsbs_b_odd_bounded * fsbs_b_odd_bounded + 1 = p * fsbs_k_odd_bounded) /\ (exists fsbs_lt_gap_odd_bounded_multiplier. fsbs_lt_gap_odd_bounded_multiplier + S (fsbs_k_odd_bounded) = (p))))

Proof neighborhood

Direct theorem prerequisites

FS000K four_square_odd_prime_half_coordinate_seed FS000M four_square_odd_prime_half_seed_norm_strict FS001O four_square_descent_norm_bound_forces_smaller_multiplier

Direct theorem dependents

FS000O four_square_non_two_prime_bounded_modular_seed

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

35 script commands · 8 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 (3)
four square odd prime half coordinate seedfour square odd prime half seed norm strictfour square descent norm bound forces smaller multiplier
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro hodd
  4. L4
    intro hprime
02Establish hseedL5–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square odd prime half coordinate seed.

  1. L5
    have hseed : ∃ a. ∃ b. ∃ k. a · a + b · b + 1 = p · k ∧ (Le(a,h) ∧ Le(b,h))Definitions: Le(a,h)Le(b,h)Original native command in the exact edition
  2. L6
    specialize four_square_odd_prime_half_coordinate_seed p
  3. L7
    specialize four_square_odd_prime_half_coordinate_seed h
  4. L8
    apply four_square_odd_prime_half_coordinate_seed
  5. L9
    exact hodd
  6. L10
    exact hprime
03Separate the logical casesL11–15

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

  1. L11
    cases hseed
  2. L12
    cases hseed_witness
  3. L13
    cases hseed_witness_witness
  4. L14
    cases hseed_witness_witness_witness
  5. L15
    cases hseed_witness_witness_witness_right
04Construct an explicit witnessL16–18

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

  1. L16
    exists x
  2. L17
    exists x1
  3. L18
    exists x2
05Separate the logical casesL19–19

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

  1. L19
    split
06Use earlier factsL20–24

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

  1. L20
    exact hseed_witness_witness_witness_left
  2. L21
    specialize four_square_descent_norm_bound_forces_smaller_multiplier p
  3. L22
    specialize four_square_descent_norm_bound_forces_smaller_multiplier x2
  4. L23
    specialize four_square_descent_norm_bound_forces_smaller_multiplier (x * x + x1 * x1 + 1)
  5. L24
    apply four_square_descent_norm_bound_forces_smaller_multiplier
07Calculate and transport equalitiesL25–25

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

  1. L25
    symm
08Use earlier factsL26–35

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

  1. L26
    exact hseed_witness_witness_witness_left
  2. L27
    specialize four_square_odd_prime_half_seed_norm_strict p
  3. L28
    specialize four_square_odd_prime_half_seed_norm_strict h
  4. L29
    specialize four_square_odd_prime_half_seed_norm_strict x
  5. L30
    specialize four_square_odd_prime_half_seed_norm_strict x1
  6. L31
    apply four_square_odd_prime_half_seed_norm_strict
  7. L32
    exact hodd
  8. L33
    exact hprime
  9. L34
    exact hseed_witness_witness_witness_right_left
  10. L35
    exact hseed_witness_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro hodd
  4. 0004intro hprime
  5. 0005have hseed : ∃ a. ∃ b. ∃ k. a · a + b · b + 1 = p · k ∧ (Le(a,h) ∧ Le(b,h))
    Exact native replay linehave hseed : exists a b k. ((a * a + b * b + 1 = p * k) /\ ((exists gap. gap + a = h) /\ (exists gap. gap + b = h)))
  6. 0006specialize four_square_odd_prime_half_coordinate_seed p
  7. 0007specialize four_square_odd_prime_half_coordinate_seed h
  8. 0008apply four_square_odd_prime_half_coordinate_seed
  9. 0009exact hodd
  10. 0010exact hprime
  11. 0011cases hseed
  12. 0012cases hseed_witness
  13. 0013cases hseed_witness_witness
  14. 0014cases hseed_witness_witness_witness
  15. 0015cases hseed_witness_witness_witness_right
  16. 0016exists x
  17. 0017exists x1
  18. 0018exists x2
  19. 0019split
  20. 0020exact hseed_witness_witness_witness_left
  21. 0021specialize four_square_descent_norm_bound_forces_smaller_multiplier p
  22. 0022specialize four_square_descent_norm_bound_forces_smaller_multiplier x2
  23. 0023specialize four_square_descent_norm_bound_forces_smaller_multiplier (x * x + x1 * x1 + 1)
  24. 0024apply four_square_descent_norm_bound_forces_smaller_multiplier
  25. 0025symm
  26. 0026exact hseed_witness_witness_witness_left
  27. 0027specialize four_square_odd_prime_half_seed_norm_strict p
  28. 0028specialize four_square_odd_prime_half_seed_norm_strict h
  29. 0029specialize four_square_odd_prime_half_seed_norm_strict x
  30. 0030specialize four_square_odd_prime_half_seed_norm_strict x1
  31. 0031apply four_square_odd_prime_half_seed_norm_strict
  32. 0032exact hodd
  33. 0033exact hprime
  34. 0034exact hseed_witness_witness_witness_right_left
  35. 0035exact hseed_witness_witness_witness_right_right