TS000F

prime_floor_bounded_coordinate_square_strict

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every coordinate at most the floor square root of a prime has square below that 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.

Exact expanded first-order arithmetic statement

forall p s a. ((~(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_coordinate_bound. ftsp_weak_coordinate_bound + (a) = s) -> (exists ftsp_gap_coordinate_square_bound. ftsp_gap_coordinate_square_bound + S (a * a) = p)

Constructive proof overview

Generated structural guide

Every coordinate at most the floor square root of a prime has square below that prime.

The unchanged tactic script uses 3 declared prerequisites and contains 23 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

TS000D natural_square_monotone_expanded TS000E prime_floor_square_strictly_below_prime lt_of_le_of_lt Stable theorem; checked-use authorized

Direct dependents

TS000H prime_floor_bounded_two_square_norm_below_double

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

23 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.

Named ingredients (2)
natural square monotone expandedprime floor square strictly below prime
01Fix variables and assumptionsL1–6

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 hprime
  5. L5
    intro hfloor
  6. L6
    intro hbound
02Establish hsquareL7–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply natural square monotone expanded.

  1. L7
    have hsquare : exists k. k + a * a = s * s
  2. L8
    specialize natural_square_monotone_expanded a
  3. L9
    specialize natural_square_monotone_expanded s
  4. L10
    apply natural_square_monotone_expanded
  5. L11
    exact hbound
03Establish hprimeboundL12–21

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

  1. L12
    have hprimebound : exists k. k + S (s * s) = p
  2. L13
    specialize prime_floor_square_strictly_below_prime p
  3. L14
    specialize prime_floor_square_strictly_below_prime s
  4. L15
    apply prime_floor_square_strictly_below_prime
  5. L16
    exact hprime
  6. L17
    exact hfloor
  7. L18
    specialize lt_of_le_of_lt (a * a)
  8. L19
    specialize lt_of_le_of_lt (s * s)
  9. L20
    specialize lt_of_le_of_lt p
  10. L21
    apply lt_of_le_of_lt
04Use earlier factsL22–23

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

  1. L22
    exact hsquare
  2. L23
    exact hprimebound

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro p
  2. 0002intro s
  3. 0003intro a
  4. 0004intro hprime
  5. 0005intro hfloor
  6. 0006intro hbound
  7. 0007have hsquare : exists k. k + a * a = s * s
  8. 0008specialize natural_square_monotone_expanded a
  9. 0009specialize natural_square_monotone_expanded s
  10. 0010apply natural_square_monotone_expanded
  11. 0011exact hbound
  12. 0012have hprimebound : exists k. k + S (s * s) = p
  13. 0013specialize prime_floor_square_strictly_below_prime p
  14. 0014specialize prime_floor_square_strictly_below_prime s
  15. 0015apply prime_floor_square_strictly_below_prime
  16. 0016exact hprime
  17. 0017exact hfloor
  18. 0018specialize lt_of_le_of_lt (a * a)
  19. 0019specialize lt_of_le_of_lt (s * s)
  20. 0020specialize lt_of_le_of_lt p
  21. 0021apply lt_of_le_of_lt
  22. 0022exact hsquare
  23. 0023exact hprimebound