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 → ¬k = 0 ∧ Le(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 -> (~(k = 0) /\ (exists bcf_le_gap_bdhpb_result. bcf_le_gap_bdhpb_result + (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 kL2–3
03Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
exfalso
04Use earlier factsL5–5
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L5
apply PA1
05Calculate and transport equalitiesL6–6
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L6
trans 2 * 0
06Use earlier factsL7–7
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L7
exact heq
07Calculate and transport equalitiesL8–8
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L8
norm_num
08Fix variables and assumptionsL9–9
Work with arbitrary variables or the premises of the current implication.
- L9
intro heq
09Separate the logical casesL10–10
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L10
split
10Fix variables and assumptionsL11–11
Work with arbitrary variables or the premises of the current implication.
- L11
intro hk
11Use earlier factsL12–13
12Construct an explicit witnessL14–14
Supply the displayed value, then prove that it has the required property.
- L14
exists k
13Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
apply PA2
14Calculate and transport equalitiesL16–18
15Use earlier factsL19–19
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L19
exact heq