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. n = 0 \/ exists k. n = S kStructural proof guide
Every natural is either zero or the successor of a natural.
Direct prerequisites: none. The authored body proceeds by structural induction (1).
Proof neighborhood
Direct dependencies
Direct dependents
BT001C le_eq_or_lt BT001O division_remainder_succ BT001P division_remainder_exists BT001U division_remainder_unique BT001W multiple_has_zero_remainder BT002F multiple_antisymm BT004M bounded_common_multiple_step BT006N prime_three BT00R1 ceil_div_six_total BT00RL prime_power_valuation_one_zero BT00T2 beta_pascal_zero_row_extend BT00T4 beta_pascal_row_step_extend BT00T6 beta_pascal_table_prefix_extend BT00TT choose_weighted_vertical BT00VV primorial_le_four_pow_bounded BT00YA prime_square_tail_of_two_three_range BT00YC double_quotient_carry_prefix_entries_zero BT00YD central_binom_prime_valuation_zero_of_exact_double_quotientsFormal 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
02Separate the logical casesL2–2
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L2
left
03Calculate and transport equalitiesL3–3
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L3
refl
04Separate the logical casesL4–4
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L4
right
05Construct an explicit witnessL5–5
Supply the displayed value, then prove that it has the required property.
- L5
exists n
06Calculate and transport equalitiesL6–6
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L6
refl