PA001B

le_zero

Stable checked-use theorem · independently closed

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

Generated structural guide

Only zero is less than or equal to zero.

Use the direct prerequisites add_eq_zero_right as previously established PA formulas.

The proof proceeds by case analysis (1).

Referenced ingredients

PA0004 add_eq_zero_right

Proof neighborhood

Direct dependencies

PA0004 add_eq_zero_right

Direct dependents

PA001K gcd_balanced_bezout_exists_up_to PA0035 gcd_exists_up_to PA00DX eisenstein_initial_segment_bit_count_functional

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

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