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 n. (exists ge_gap_natural_square. ge_gap_natural_square + (n) = (n*n))Constructive proof overview
Generated structural guide
Every natural is at most its square, including zero, by constructive equality decision.
The unchanged tactic script uses 2 declared prerequisites and contains 12 exact native proof lines.
Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
eq_decidable Stable theorem; checked-use authorized le_scaled_nonzero Stable theorem; checked-use authorizedDirect dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
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 assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro n
02Establish hnL2–5
03Separate the logical casesL6–6
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L6
cases hn
04Construct an explicit witnessL7–7
Supply the displayed value, then prove that it has the required property.
- L7
exists 0
05Calculate and transport equalitiesL8–8
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L8
simp [hn_left]