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.
Exact expanded PA statement
forall n. exists k. n = 2 * k \/ n = 2 * k + 1Structural proof guide
Every natural has a constructive even-or-odd witness.
Direct prerequisites: none. The authored body proceeds by structural induction (1), case analysis (2), equality transport (2), closed numeral normalization (1).
Proof neighborhood
Direct dependencies
Direct dependents
Formal native tactic body
Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.
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.
01Induction on nL1–1
Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.
- L1
induction n
02Construct an explicit witnessL2–2
Supply the displayed value, then prove that it has the required property.
- L2
exists 0
03Separate the logical casesL3–3
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L3
left
04Calculate and transport equalitiesL4–4
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L4
norm_num
05Separate the logical casesL5–6
06Construct an explicit witnessL7–7
Supply the displayed value, then prove that it has the required property.
- L7
exists x
07Separate the logical casesL8–8
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L8
right
08Calculate and transport equalitiesL9–10
09Construct an explicit witnessL11–11
Supply the displayed value, then prove that it has the required property.
- L11
exists x + 1
10Separate the logical casesL12–12
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L12
left