TS000B · theorem body

bounded_divisible_two_square_norm_equals_prime

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

A positive two-square norm divisible by p and strictly below 2p is exactly p.

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. ∀ a. ∀ b. Lt(0,a · a + b · b) → Dvd(p,a · a + b · b) → Lt(a · a + b · b,p + p) → p = a · a + b · b

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 PD0003 Dvd

In local proof propositions

none
Exact expanded first-order statement
forall p a b. (exists fts_gap_norm_positive. fts_gap_norm_positive + S (0) = a * a + b * b) -> (exists fts_factor_norm_divisible. a * a + b * b = p * fts_factor_norm_divisible) -> (exists fts_gap_norm_upper. fts_gap_norm_upper + S (a * a + b * b) = p + p) -> p = a * a + b * b

Proof neighborhood

Direct theorem prerequisites

TS000A positive_multiple_below_twice_equals_base

Direct theorem dependents

TS000K prime_floor_bounded_divisible_norm_represents_prime

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

15 script commands · 2 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)
positive multiple below twice equals base
01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro hpositive
  5. L5
    intro hdivisible
  6. L6
    intro hupper
02Establish hequalityL7–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply positive multiple below twice equals base.

  1. L7
    have hequality : a * a + b * b = p
  2. L8
    specialize positive_multiple_below_twice_equals_base p
  3. L9
    specialize positive_multiple_below_twice_equals_base (a * a + b * b)
  4. L10
    apply positive_multiple_below_twice_equals_base
  5. L11
    exact hpositive
  6. L12
    exact hdivisible
  7. L13
    exact hupper
  8. L14
    symm
  9. L15
    exact hequality

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hpositive
  5. 0005intro hdivisible
  6. 0006intro hupper
  7. 0007have hequality : a * a + b * b = p
  8. 0008specialize positive_multiple_below_twice_equals_base p
  9. 0009specialize positive_multiple_below_twice_equals_base (a * a + b * b)
  10. 0010apply positive_multiple_below_twice_equals_base
  11. 0011exact hpositive
  12. 0012exact hdivisible
  13. 0013exact hupper
  14. 0014symm
  15. 0015exact hequality