FS000M · theorem body

four_square_odd_prime_half_seed_norm_strict

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

The actual half-range seed norm a²+b²+1 is strictly smaller than the square of its odd prime modulus.

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. ∀ a. ∀ b. p = 2 · h + 1 → Prime(p) → Le(a,h) → Le(b,h) → Lt(a · a + b · b + 1,p · 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

PD0001 Le PD0002 Lt PD0004 Prime

In local proof propositions

PD0002 Lt
Exact expanded first-order statement
forall p h a b. 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_le_gap_strict_first. fsbs_le_gap_strict_first + (a) = (h)) -> (exists fsbs_le_gap_strict_second. fsbs_le_gap_strict_second + (b) = (h)) -> (exists fsbs_lt_gap_strict_norm. fsbs_lt_gap_strict_norm + S (a * a + b * b + 1) = (p * p))

Proof neighborhood

Direct theorem prerequisites

FS000L four_square_odd_prime_half_positive FS001X four_square_descent_odd_half_norm_strict

Direct theorem dependents

FS000N four_square_odd_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

32 script commands · 6 reading checkpoints · 3 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)
four square odd prime half positivefour square descent odd half norm strict
01Fix variables and assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro a
  4. L4
    intro b
  5. L5
    intro hodd
  6. L6
    intro hprime
  7. L7
    intro ha
  8. L8
    intro hb
02Establish hpositiveL9–14

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

  1. L9
    have hpositive : Lt(0,h)Definitions: Lt(0,h)Original native command in the exact edition
  2. L10
    specialize four_square_odd_prime_half_positive p
  3. L11
    specialize four_square_odd_prime_half_positive h
  4. L12
    apply four_square_odd_prime_half_positive
  5. L13
    exact hodd
  6. L14
    exact hprime
03Establish hnormL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square descent odd half norm strict.

  1. L15
    have hnorm : Lt(a · a + b · b + 1 · 1 + 0 · 0,(2 · h + 1) · (2 · h + 1))Definitions: Lt(a · a + b · b + 1 · 1 + 0 · 0,(2 · h + 1) · (2 · h + 1))Original native command in the exact edition
  2. L16
    specialize four_square_descent_odd_half_norm_strict h
  3. L17
    specialize four_square_descent_odd_half_norm_strict a
  4. L18
    specialize four_square_descent_odd_half_norm_strict b
  5. L19
    specialize four_square_descent_odd_half_norm_strict 1
  6. L20
    specialize four_square_descent_odd_half_norm_strict 0
  7. L21
    apply four_square_descent_odd_half_norm_strict
  8. L22
    exact ha
  9. L23
    exact hb
  10. L24
    exact hpositive
04Construct an explicit witnessL25–25

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

  1. L25
    exists h
05Calculate and transport equalitiesL26–26

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

  1. L26
    simp
06Establish hshapeL27–32

Establish this local claim before using it. It is not an additional assumption.

  1. L27
    have hshape : a * a + b * b + 1 * 1 + 0 * 0 = a * a + b * b + 1
  2. L28
    simp
  3. L29
    rewrite hshape at hnorm
  4. L30
    rewrite hodd
  5. L31
    rewrite hodd
  6. L32
    exact hnorm

Library-wide reading audit

Original defined command ledger · 32 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro a
  4. 0004intro b
  5. 0005intro hodd
  6. 0006intro hprime
  7. 0007intro ha
  8. 0008intro hb
  9. 0009have hpositive : Lt(0,h)
    Exact native replay linehave hpositive : exists gap. gap + 1 = h
  10. 0010specialize four_square_odd_prime_half_positive p
  11. 0011specialize four_square_odd_prime_half_positive h
  12. 0012apply four_square_odd_prime_half_positive
  13. 0013exact hodd
  14. 0014exact hprime
  15. 0015have hnorm : Lt(a · a + b · b + 1 · 1 + 0 · 0,(2 · h + 1) · (2 · h + 1))
    Exact native replay linehave hnorm : exists gap. gap + S (a * a + b * b + 1 * 1 + 0 * 0) = (2 * h + 1) * (2 * h + 1)
  16. 0016specialize four_square_descent_odd_half_norm_strict h
  17. 0017specialize four_square_descent_odd_half_norm_strict a
  18. 0018specialize four_square_descent_odd_half_norm_strict b
  19. 0019specialize four_square_descent_odd_half_norm_strict 1
  20. 0020specialize four_square_descent_odd_half_norm_strict 0
  21. 0021apply four_square_descent_odd_half_norm_strict
  22. 0022exact ha
  23. 0023exact hb
  24. 0024exact hpositive
  25. 0025exists h
  26. 0026simp
  27. 0027have hshape : a * a + b * b + 1 * 1 + 0 * 0 = a * a + b * b + 1
  28. 0028simp
  29. 0029rewrite hshape at hnorm
  30. 0030rewrite hodd
  31. 0031rewrite hodd
  32. 0032exact hnorm