BT000Y

le_zero

Stable checked-use theorem · independently kernel verified

Only zero is less than or equal to zero.

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 -> n = 0

Structural proof guide

Only zero is less than or equal to zero.

Direct prerequisites: add_eq_zero_right. The authored body proceeds by case analysis (1).

Proof neighborhood

Direct dependencies

BT000L add_eq_zero_right

Direct dependents

BT002U gcd_exists_up_to BT0034 gcd_balanced_bezout_exists_up_to BT003E factor_search_up_to BT003K prime_divisor_exists_up_to BT0097 lt_three_cases BT00JD eisenstein_initial_segment_bit_count_functional BT00PS bounded_prime_interval_search BT00Q5 bounded_power_valuation_search BT00TM choose_positive BT00VV primorial_le_four_pow_bounded

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

5 script commands · 3 reading checkpoints · 0 local claims

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.

Named ingredients (1)
add eq zero right
01Fix variables and assumptionsL1–2

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
  2. L2
    intro h
02Separate the logical casesL3–3

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L3
    cases h
03Use earlier factsL4–5

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L4
    apply add_eq_zero_right
  2. L5
    exact h_witness

Library-wide reading audit

Original exact command ledger · 5 lines
  1. 0001intro n
  2. 0002intro h
  3. 0003cases h
  4. 0004apply add_eq_zero_right
  5. 0005exact h_witness