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
∀ N. ∀ C. ∀ B. (∀ x. (∃ y. S x + S y = S N) → Dvd(S x,C)) → ∀ x. (∃ y. S x + S y = S N) → Dvd(S x,C · B)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
2 occurrences
In local proof propositions
1 occurrences
Exact expanded native-PA statement
forall N C B. (forall t. (exists h. S t + S h = S N) -> exists q. C = S t * q) -> forall t. (exists h. S t + S h = S N) -> exists q. C * B = S t * qProof 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.
Named ingredients (1)
01Fix variables and assumptionsL1–6
02Establish htCL7–15
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hcm.
Original defined command ledger · 15 lines
- 0001
intro N - 0002
intro C - 0003
intro B - 0004
intro hcm - 0005
intro t - 0006
intro ht - 0007
have htC : Dvd(S t,C)Exact native replay line
have htC : exists q. C = S t * q - 0008
specialize hcm t - 0009
apply hcm - 0010
exact ht - 0011
specialize multiple_mul_right (S t) - 0012
specialize multiple_mul_right C - 0013
specialize multiple_mul_right B - 0014
apply multiple_mul_right - 0015
exact htC