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
forall m n. m * n + n * m = 2 * (m * n)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
In local proof propositions
Exact expanded first-order statement
forall m n. m * n + n * m = 2 * (m * n)Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
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
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 assumptionsL1–2
02Calculate and transport equalitiesL3–5
03Use earlier factsL6–6
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L6
apply mul_comm
04Calculate and transport equalitiesL7–8
05Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
apply mul_succ_left
06Calculate and transport equalitiesL10–10
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L10
congr
07Use earlier factsL11–11
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L11
apply one_mul
08Calculate and transport equalitiesL12–12
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L12
refl