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
∀ c. ∀ b. ∀ q. ∀ r. Dvd(c,b) → Dvd(c,r) → Dvd(c,b · q + r)Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
3 occurrences
In local proof propositions
0 occurrences
Exact expanded native-PA statement
forall c b q r. (exists u. b = c * u) -> (exists v. r = c * v) -> exists w. b * q + r = c * wProof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
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 assumptionsL1–6
02Separate the logical casesL7–8
03Construct an explicit witnessL9–9
Supply the displayed value, then prove that it has the required property.
- L9
exists x * q + x1
04Calculate and transport equalitiesL10–13
05Use earlier factsL14–14
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L14
apply mul_assoc
06Calculate and transport equalitiesL15–16
07Use earlier factsL17–17
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L17
apply mul_add