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
∀ a. ∀ b. Lt(0,a) → Le(b,a · 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 a b. (exists bpo_gap_left_factor. bpo_gap_left_factor + (1) = (a)) -> (exists bpo_gap_left_result. bpo_gap_left_result + (b) = (a * b))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 (2)
01Fix variables and assumptionsL1–3
02Establish hscaledL4–12
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul right.
Original defined command ledger · 12 lines
- 0001
intro a - 0002
intro b - 0003
intro ha - 0004
have hscaled : Le(1 · b,a · b)Exact native replay line
have hscaled : exists k. k + 1 * b = a * b - 0005
specialize mul_le_mul_right 1 - 0006
specialize mul_le_mul_right a - 0007
specialize mul_le_mul_right b - 0008
apply mul_le_mul_right - 0009
exact ha - 0010
specialize one_mul b - 0011
rewrite one_mul at hscaled - 0012
exact hscaled