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
∀ m. ∀ n. Le(n,m) → ∃ x. m · m = n · n + xEvery 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. (exists gap. gap + n = m) -> exists d. m * m = n * n + dProof 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 assumptionsL1–3
02Establish hsquareL4–7
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul.
- L4
have hsquare : Le(n · n,m · m)Definitions: Le(n · n,m · m)Original native command in the exact edition - L5
apply mul_le_mul - L6
exact hbound - L7
exact hbound
03Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
cases hsquare
04Construct an explicit witnessL9–9
Supply the displayed value, then prove that it has the required property.
- L9
exists x