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. ∀ i. Le(c,S (S i · c))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
1 occurrences
In local proof propositions
1 occurrences
Exact expanded native-PA statement
forall c i. exists h. h + c = S ((S i) * c)Proof 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 (3)
01Fix variables and assumptionsL1–2
02Establish hproductL3–12
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le succ.
03Use earlier factsL13–13
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L13
exact hproduct
Original defined command ledger · 13 lines
- 0001
intro c - 0002
intro i - 0003
have hproduct : Le(c,S i · c)Exact native replay line
have hproduct : exists h. h + c = S i * c - 0004
specialize mul_succ_left i - 0005
specialize mul_succ_left c - 0006
rewrite mul_succ_left - 0007
specialize le_add_left c - 0008
specialize le_add_left (i * c) - 0009
exact le_add_left - 0010
specialize le_succ c - 0011
specialize le_succ (S i * c) - 0012
apply le_succ - 0013
exact hproduct