TS000K · theorem body

prime_floor_bounded_divisible_norm_represents_prime

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

A positive divisible norm with floor-square-bounded coordinates is already an exact prime representation.

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. ∀ s. ∀ a. ∀ b. Prime(p) → FloorSqrt(p,s) → Le(a,s) → Le(b,s) → Lt(0,a · a + b · b) → Dvd(p,a · a + b · b) → 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

PD0001 Le PD0002 Lt PD0003 Dvd PD0004 Prime PD0051 FloorSqrt

In local proof propositions

PD0002 Lt
Exact expanded first-order statement
forall p s a b. ((~(p = 1) /\ forall frm_prime_left_ftsp_prime frm_prime_right_ftsp_prime. p = frm_prime_left_ftsp_prime * frm_prime_right_ftsp_prime -> frm_prime_left_ftsp_prime = 1 \/ frm_prime_right_ftsp_prime = 1)) -> (((exists bcs_sqrt_lower_gap_ftsp_prime_floor. bcs_sqrt_lower_gap_ftsp_prime_floor + (s) * (s) = (p)) /\ exists bcs_sqrt_upper_gap_ftsp_prime_floor. bcs_sqrt_upper_gap_ftsp_prime_floor + S (p) = S (s) * S (s))) -> (exists ftsp_weak_representation_first. ftsp_weak_representation_first + (a) = s) -> (exists ftsp_weak_representation_second. ftsp_weak_representation_second + (b) = s) -> (exists ftsp_gap_representation_positive. ftsp_gap_representation_positive + S (0) = a * a + b * b) -> (exists k. a * a + b * b = p * k) -> p = a * a + b * b

Proof neighborhood

Direct theorem prerequisites

TS000H prime_floor_bounded_two_square_norm_below_double TS000B bounded_divisible_two_square_norm_equals_prime

Direct theorem dependents

TS001M prime_floor_decoded_affine_collision_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

27 script commands · 3 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 (2)
prime floor bounded two square norm below doublebounded divisible two square norm equals prime
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro s
  3. L3
    intro a
  4. L4
    intro b
  5. L5
    intro hprime
  6. L6
    intro hfloor
  7. L7
    intro hfirst
  8. L8
    intro hsecond
  9. L9
    intro hpositive
  10. L10
    intro hdivisible
02Establish hupperL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime floor bounded two square norm below double.

  1. L11
    have hupper : Lt(a · a + b · b,p + p)Definitions: Lt(a · a + b · b,p + p)Original native command in the exact edition
  2. L12
    specialize prime_floor_bounded_two_square_norm_below_double p
  3. L13
    specialize prime_floor_bounded_two_square_norm_below_double s
  4. L14
    specialize prime_floor_bounded_two_square_norm_below_double a
  5. L15
    specialize prime_floor_bounded_two_square_norm_below_double b
  6. L16
    apply prime_floor_bounded_two_square_norm_below_double
  7. L17
    exact hprime
  8. L18
    exact hfloor
  9. L19
    exact hfirst
  10. L20
    exact hsecond
03Use earlier factsL21–27

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

  1. L21
    specialize bounded_divisible_two_square_norm_equals_prime p
  2. L22
    specialize bounded_divisible_two_square_norm_equals_prime a
  3. L23
    specialize bounded_divisible_two_square_norm_equals_prime b
  4. L24
    apply bounded_divisible_two_square_norm_equals_prime
  5. L25
    exact hpositive
  6. L26
    exact hdivisible
  7. L27
    exact hupper

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro p
  2. 0002intro s
  3. 0003intro a
  4. 0004intro b
  5. 0005intro hprime
  6. 0006intro hfloor
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009intro hpositive
  10. 0010intro hdivisible
  11. 0011have hupper : Lt(a · a + b · b,p + p)
    Exact native replay linehave hupper : exists k. k + S (a * a + b * b) = p + p
  12. 0012specialize prime_floor_bounded_two_square_norm_below_double p
  13. 0013specialize prime_floor_bounded_two_square_norm_below_double s
  14. 0014specialize prime_floor_bounded_two_square_norm_below_double a
  15. 0015specialize prime_floor_bounded_two_square_norm_below_double b
  16. 0016apply prime_floor_bounded_two_square_norm_below_double
  17. 0017exact hprime
  18. 0018exact hfloor
  19. 0019exact hfirst
  20. 0020exact hsecond
  21. 0021specialize bounded_divisible_two_square_norm_equals_prime p
  22. 0022specialize bounded_divisible_two_square_norm_equals_prime a
  23. 0023specialize bounded_divisible_two_square_norm_equals_prime b
  24. 0024apply bounded_divisible_two_square_norm_equals_prime
  25. 0025exact hpositive
  26. 0026exact hdivisible
  27. 0027exact hupper