TS000J · theorem body

floor_square_oversized_grid_exists

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

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.

Statement with defined notation

∀ p. ∃ s. FloorSqrt(p,s) ∧ Lt(p,S s · S s)

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 PD0051 FloorSqrt

In local proof propositions

PD0051 FloorSqrt
Exact expanded first-order 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))

Proof neighborhood

Direct theorem prerequisites

floor_sqrt_total · Alpha closed TS000I floor_square_successor_grid_strictly_exceeds_input

Direct theorem dependents

none

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

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.

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)
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 : ∃ s. FloorSqrt(p,s)Definitions: FloorSqrt(p,s)Original native command in the exact edition
  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 defined command ledger · 12 lines
  1. 0001intro p
  2. 0002have hfloor : ∃ s. FloorSqrt(p,s)
    Exact native replay linehave 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