TS000J

floor_square_oversized_grid_exists

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Every natural admits an explicitly witnessed floor-square grid with more cells than that natural.

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. exists s. ((((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_existing_grid_overflow. ftsp_gap_existing_grid_overflow + S (p) = S s * S s))

Constructive proof overview

Generated structural guide

Every natural admits an explicitly witnessed floor-square grid with more cells than that natural.

The unchanged tactic script uses 2 declared prerequisites and contains 12 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

floor_sqrt_total Alpha theorem; checked-use authorized TS000I floor_square_successor_grid_strictly_exceeds_input

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

12 script commands · 6 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.

Named ingredients (1)
floor square successor grid strictly exceeds input
01Fix variables and assumptionsL1–1

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

  1. L1
    intro p
02Establish hfloorL2–4

Establish this local claim before using it. It is not an additional assumption.

  1. L2
    have hfloor : exists s. (((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)))
  2. L3
    specialize floor_sqrt_total p
  3. L4
    exact floor_sqrt_total
03Separate the logical casesL5–5

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

  1. L5
    cases hfloor
04Construct an explicit witnessL6–6

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists x
05Separate the logical casesL7–7

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

  1. L7
    split
06Use earlier factsL8–12

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

  1. L8
    exact hfloor_witness
  2. L9
    specialize floor_square_successor_grid_strictly_exceeds_input p
  3. L10
    specialize floor_square_successor_grid_strictly_exceeds_input x
  4. L11
    apply floor_square_successor_grid_strictly_exceeds_input
  5. L12
    exact hfloor_witness

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro p
  2. 0002have hfloor : exists s. (((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)))
  3. 0003specialize floor_sqrt_total p
  4. 0004exact floor_sqrt_total
  5. 0005cases hfloor
  6. 0006exists x
  7. 0007split
  8. 0008exact hfloor_witness
  9. 0009specialize floor_square_successor_grid_strictly_exceeds_input p
  10. 0010specialize floor_square_successor_grid_strictly_exceeds_input x
  11. 0011apply floor_square_successor_grid_strictly_exceeds_input
  12. 0012exact hfloor_witness