TS000E · theorem body

prime_floor_square_strictly_below_prime

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

For a prime, its floor-square lower endpoint is strictly smaller than 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. Prime(p) → FloorSqrt(p,s) → Lt(s · s,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

PD0002 Lt PD0004 Prime PD0051 FloorSqrt

In local proof propositions

PD0002 Lt
Exact expanded first-order statement
forall p s. ((~(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_gap_prime_floor_strict. ftsp_gap_prime_floor_strict + S (s * s) = p)

Proof neighborhood

Direct theorem prerequisites

TS000C prime_is_not_natural_square le_eq_or_lt · Stable closed

Direct theorem dependents

TS000F prime_floor_bounded_coordinate_square_strict

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

19 script commands · 7 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 (1)
prime is not natural square
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro s
  3. L3
    intro hprime
  4. L4
    intro hfloor
02Separate the logical casesL5–5

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L5
    cases hfloor
03Establish hsplitL6–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le eq or lt.

  1. L6
    have hsplit : s · s = p ∨ Lt(s · s,p)Definitions: Lt(s · s,p)Original native command in the exact edition
  2. L7
    specialize le_eq_or_lt (s * s)
  3. L8
    specialize le_eq_or_lt p
  4. L9
    apply le_eq_or_lt
  5. L10
    exact hfloor_left
04Separate the logical casesL11–12

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L11
    cases hsplit
  2. L12
    exfalso
05Use earlier factsL13–16

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

  1. L13
    specialize prime_is_not_natural_square p
  2. L14
    specialize prime_is_not_natural_square s
  3. L15
    apply prime_is_not_natural_square
  4. L16
    exact hprime
06Calculate and transport equalitiesL17–17

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L17
    symm
07Use earlier factsL18–19

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

  1. L18
    exact hsplit_left
  2. L19
    exact hsplit_right

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro s
  3. 0003intro hprime
  4. 0004intro hfloor
  5. 0005cases hfloor
  6. 0006have hsplit : s · s = p ∨ Lt(s · s,p)
    Exact native replay linehave hsplit : s * s = p \/ exists k. k + S (s * s) = p
  7. 0007specialize le_eq_or_lt (s * s)
  8. 0008specialize le_eq_or_lt p
  9. 0009apply le_eq_or_lt
  10. 0010exact hfloor_left
  11. 0011cases hsplit
  12. 0012exfalso
  13. 0013specialize prime_is_not_natural_square p
  14. 0014specialize prime_is_not_natural_square s
  15. 0015apply prime_is_not_natural_square
  16. 0016exact hprime
  17. 0017symm
  18. 0018exact hsplit_left
  19. 0019exact hsplit_right