GI0035

gaussian_nonzero_natural_positive

Every nonzero natural has an explicit strict-positive gap witness.

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 natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.

Exact theorem in conservative defined notation

∀ n. ¬n = 0 → Lt(0,n)

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

nonzero_is_succ · checked external prerequisite
Original expanded first-order statement 
forall n. ~(n = 0) -> (exists ge_gap_positive. ge_gap_positive + S (0) = (n))

Complete tactic proof in conservative notation

All 10 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

10 script commands · 6 reading checkpoints · 1 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 n
  2. L2
    intro hn
02Use earlier factsL3–3

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

  1. L3
    specialize nonzero_is_succ n
03Establish hsuccessorL4–6

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L4
    have hsuccessor : exists k. n = S k
  2. L5
    apply nonzero_is_succ
  3. L6
    exact hn
04Separate the logical casesL7–7

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

  1. L7
    cases hsuccessor
05Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists x
06Calculate and transport equalitiesL9–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    rewrite hsuccessor_witness
  2. L10
    simp

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003specialize nonzero_is_succ n
  4. 0004have hsuccessor : exists k. n = S k
  5. 0005apply nonzero_is_succ
  6. 0006exact hn
  7. 0007cases hsuccessor
  8. 0008exists x
  9. 0009rewrite hsuccessor_witness
  10. 0010simp