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. ∀ k. S n = 2 · k + 1 → (∃ x. k = S x) → Lt(k,n)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
0 occurrences
Exact expanded native-PA statement
forall n k. S n = 2 * k + 1 -> (exists h. k = S h) -> (exists bcf_le_gap_boppb_result. bcf_le_gap_boppb_result + (S k) = n)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.
01Fix variables and assumptionsL1–1
Work with arbitrary variables or the premises of the current implication.
- L1
intro n
02Induction on kL2–4
03Separate the logical casesL5–6
04Use earlier factsL7–7
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L7
apply PA1
05Calculate and transport equalitiesL8–8
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L8
symm
06Use earlier factsL9–9
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L9
exact hpositive_witness
07Fix variables and assumptionsL10–11
08Construct an explicit witnessL12–12
Supply the displayed value, then prove that it has the required property.
- L12
exists k
09Use earlier factsL13–13
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L13
apply PA2
10Calculate and transport equalitiesL14–16
11Use earlier factsL17–17
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L17
exact heq