TS000H · theorem body

prime_floor_bounded_two_square_norm_below_double

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

Any two coordinates bounded by the prime floor square root have norm strictly below twice the prime.

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(a · a + b · b,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 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_norm_first_bound. ftsp_weak_norm_first_bound + (a) = s) -> (exists ftsp_weak_norm_second_bound. ftsp_weak_norm_second_bound + (b) = s) -> (exists ftsp_gap_norm_upper. ftsp_gap_norm_upper + S (a * a + b * b) = p + p)

Proof neighborhood

Direct theorem prerequisites

TS000F prime_floor_bounded_coordinate_square_strict TS000G two_strict_values_sum_below_double

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

30 script commands · 4 reading checkpoints · 2 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 coordinate square stricttwo strict values sum below double
01Fix variables and assumptionsL1–8

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
02Establish hfirstsquareL9–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime floor bounded coordinate square strict.

  1. L9
    have hfirstsquare : Lt(a · a,p)Definitions: Lt(a · a,p)Original native command in the exact edition
  2. L10
    specialize prime_floor_bounded_coordinate_square_strict p
  3. L11
    specialize prime_floor_bounded_coordinate_square_strict s
  4. L12
    specialize prime_floor_bounded_coordinate_square_strict a
  5. L13
    apply prime_floor_bounded_coordinate_square_strict
  6. L14
    exact hprime
  7. L15
    exact hfloor
  8. L16
    exact hfirst
03Establish hsecondsquareL17–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime floor bounded coordinate square strict.

  1. L17
    have hsecondsquare : Lt(b · b,p)Definitions: Lt(b · b,p)Original native command in the exact edition
  2. L18
    specialize prime_floor_bounded_coordinate_square_strict p
  3. L19
    specialize prime_floor_bounded_coordinate_square_strict s
  4. L20
    specialize prime_floor_bounded_coordinate_square_strict b
  5. L21
    apply prime_floor_bounded_coordinate_square_strict
  6. L22
    exact hprime
  7. L23
    exact hfloor
  8. L24
    exact hsecond
  9. L25
    specialize two_strict_values_sum_below_double (a * a)
  10. L26
    specialize two_strict_values_sum_below_double (b * b)
04Use earlier factsL27–30

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

  1. L27
    specialize two_strict_values_sum_below_double p
  2. L28
    apply two_strict_values_sum_below_double
  3. L29
    exact hfirstsquare
  4. L30
    exact hsecondsquare

Library-wide reading audit

Original defined command ledger · 30 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. 0009have hfirstsquare : Lt(a · a,p)
    Exact native replay linehave hfirstsquare : exists k. k + S (a * a) = p
  10. 0010specialize prime_floor_bounded_coordinate_square_strict p
  11. 0011specialize prime_floor_bounded_coordinate_square_strict s
  12. 0012specialize prime_floor_bounded_coordinate_square_strict a
  13. 0013apply prime_floor_bounded_coordinate_square_strict
  14. 0014exact hprime
  15. 0015exact hfloor
  16. 0016exact hfirst
  17. 0017have hsecondsquare : Lt(b · b,p)
    Exact native replay linehave hsecondsquare : exists k. k + S (b * b) = p
  18. 0018specialize prime_floor_bounded_coordinate_square_strict p
  19. 0019specialize prime_floor_bounded_coordinate_square_strict s
  20. 0020specialize prime_floor_bounded_coordinate_square_strict b
  21. 0021apply prime_floor_bounded_coordinate_square_strict
  22. 0022exact hprime
  23. 0023exact hfloor
  24. 0024exact hsecond
  25. 0025specialize two_strict_values_sum_below_double (a * a)
  26. 0026specialize two_strict_values_sum_below_double (b * b)
  27. 0027specialize two_strict_values_sum_below_double p
  28. 0028apply two_strict_values_sum_below_double
  29. 0029exact hfirstsquare
  30. 0030exact hsecondsquare