AF0005

factor_permutation_below_zero_impossible

There is no natural index below zero; empty list and permutation contracts are genuinely vacuous.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

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.

The original division, gcd, cancellation, and factor-existence foundations are exposed through checked wrappers. Unordered uniqueness adds an actual bounded, injective, surjective index map matching repeated prime occurrences. The empty factor list represents one, not zero.

Exact theorem in conservative defined notation

∀ i. ¬Lt(i,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

le_zero · checked external prerequisite
Original expanded first-order statement 
forall i. (exists pfp_gap_empty. pfp_gap_empty + S (i) = (0)) -> false

Complete tactic proof in conservative notation

All 6 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

6 script commands · 2 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro i
  2. L2
    intro hi
02Use earlier factsL3–6

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

  1. L3
    apply PA1
  2. L4
    specialize le_zero (S i)
  3. L5
    apply le_zero
  4. L6
    exact hi

Library-wide reading audit

Original defined command ledger · 6 lines
  1. 0001intro i
  2. 0002intro hi
  3. 0003apply PA1
  4. 0004specialize le_zero (S i)
  5. 0005apply le_zero
  6. 0006exact hi