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) -> 1 <= nStructural proof guide
Every nonzero natural is at least one.
Direct prerequisites: none. The authored body proceeds by structural induction (1).
Proof neighborhood
Direct dependencies
Direct dependents
BT002D divisor_le_nonzero BT0055 le_scaled_nonzero BT00QF prime_power_exponent_le BT00QN prime_power_successor_cancel_cofactor BT00S0 prime_power_quotient_prefix_exists BT00W0 power_valuation_nonzero_exponent_divides_base BT00XG division_double_quotient_bit BT00Y5 central_binom_prime_power_contribution_le_double BT00Y6 central_binom_prime_square_tail_exponent_not_two_le BT00YE central_binom_prime_valuation_zero_two_thirds_range BT00YK no_bertrand_central_nonzero_valuation_factor_rangesFormal 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 nL1–2
02Separate the logical casesL3–3
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L3
exfalso
03Use earlier factsL4–4
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L4
apply h
04Calculate and transport equalitiesL5–5
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L5
refl
05Fix variables and assumptionsL6–6
Work with arbitrary variables or the premises of the current implication.
- L6
intro h
06Construct an explicit witnessL7–7
Supply the displayed value, then prove that it has the required property.
- L7
exists n
07Calculate and transport equalitiesL8–8
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L8
simp