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 a b n. ((((a) * (a)) + ((b) * (b))) = (((a) * (b)) + (n))) -> ((((b) * (b)) + ((a) * (a))) = (((b) * (a)) + (n)))Constructive proof overview
Generated structural guide
The natural-coordinate Eisenstein norm is symmetric in its two coordinates.
The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.
Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
add_comm Stable theorem; checked-use authorized mul_comm 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 assumptionsL1–4
02Calculate and transport equalitiesL5–5
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L5
trans a * a + b * b
03Use earlier factsL6–6
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L6
apply add_comm
04Calculate and transport equalitiesL7–7
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L7
trans a * b + n
05Use earlier factsL8–8
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L8
exact hnorm
06Calculate and transport equalitiesL9–9
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L9
congr
07Use earlier factsL10–10
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L10
apply mul_comm
08Calculate and transport equalitiesL11–11
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L11
refl