GI0035

gaussian_nonzero_natural_positive

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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 first-order arithmetic statement

forall n. ~(n = 0) -> (exists ge_gap_positive. ge_gap_positive + S (0) = (n))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 10 exact native proof lines.

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

Proof neighborhood

Direct dependencies

nonzero_is_succ Stable theorem; checked-use authorized

Direct dependents

GI0037 gaussian_half_double_square_strict

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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