BT00R7

floor_sqrt_strict_upper_bound

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

The floor-square graph projects its strict successor-square bound.

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 PA statement

forall x s. (((exists bcs_sqrt_lower_gap_projection. bcs_sqrt_lower_gap_projection + (s) * (s) = (x)) /\ exists bcs_sqrt_upper_gap_projection. bcs_sqrt_upper_gap_projection + S (x) = S (s) * S (s))) -> exists k. k + S x = S s * S s

Structural proof guide

The floor-square graph projects its strict successor-square bound.

Direct prerequisites: none. The authored body proceeds by case analysis (1).

Proof neighborhood

Direct dependencies

none

Direct dependents

BT00X4 bertrand_floor_power_product_le_h_from_total BT010B floor_sqrt_two_le_of_two_lt

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

5 script commands · 3 reading checkpoints · 0 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro x
  2. L2
    intro s
  3. L3
    intro h
02Separate the logical casesL4–4

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

  1. L4
    cases h
03Use earlier factsL5–5

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

  1. L5
    exact h_right

Library-wide reading audit

Original exact command ledger · 5 lines
  1. 0001intro x
  2. 0002intro s
  3. 0003intro h
  4. 0004cases h
  5. 0005exact h_right