TS000F · theorem body

prime_floor_bounded_coordinate_square_strict

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

∀ p. ∀ s. ∀ a. Prime(p) → FloorSqrt(p,s) → Le(a,s) → Lt(a · a,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

PD0001 Le PD0002 Lt
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

TS000D natural_square_monotone_expanded TS000E prime_floor_square_strictly_below_prime lt_of_le_of_lt · Stable closed

Direct theorem dependents

TS000H prime_floor_bounded_two_square_norm_below_double

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

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.

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)
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 : Le(a · a,s · s)Definitions: Le(a · a,s · s)Original native command in the exact edition
  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 : Lt(s · s,p)Definitions: Lt(s · s,p)Original native command in the exact edition
  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 defined 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 : Le(a · a,s · s)
    Exact native replay linehave 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 : Lt(s · s,p)
    Exact native replay linehave 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