TS000B

bounded_divisible_two_square_norm_equals_prime

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 15 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

TS000A positive_multiple_below_twice_equals_base

Direct dependents

TS000K prime_floor_bounded_divisible_norm_represents_prime

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

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.

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 exact 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