GF0072

gaussian_search_two_le_nonzero_not_one

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

A nonzero natural different from one is at least two, with both small boundaries explicit.

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) -> ~(n=1) -> (exists ge_gap_nonunit_norm_two. ge_gap_nonunit_norm_two + (2) = (n))

Constructive proof overview

Generated structural guide

A nonzero natural different from one is at least two, with both small boundaries explicit.

The unchanged tactic script uses 3 declared prerequisites and contains 22 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

nonzero_is_succ Stable theorem; checked-use authorized one_le_of_ne_zero Stable theorem; checked-use authorized succ_le_succ Stable theorem; checked-use authorized

Direct dependents

GF0079 gaussian_search_nonunit_norm_two

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

22 script commands · 5 reading checkpoints · 2 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 assumptionsL1–3

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

  1. L1
    intro n
  2. L2
    intro hzero
  3. L3
    intro hone
02Establish hsL4–7

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

  1. L4
    have hs : exists k. n=S k
  2. L5
    specialize nonzero_is_succ (n)
  3. L6
    apply nonzero_is_succ
  4. L7
    exact hzero
03Separate the logical casesL8–8

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

  1. L8
    cases hs
04Establish hxL9–18

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

  1. L9
    have hx : ~(x=0)
  2. L10
    intro hxzero
  3. L11
    apply hone
  4. L12
    trans S x
  5. L13
    exact hs_witness
  6. L14
    rewrite hxzero
  7. L15
    refl
  8. L16
    rewrite hs_witness
  9. L17
    specialize succ_le_succ (1)
  10. L18
    specialize succ_le_succ (x)
05Use earlier factsL19–22

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

  1. L19
    apply succ_le_succ
  2. L20
    specialize one_le_of_ne_zero (x)
  3. L21
    apply one_le_of_ne_zero
  4. L22
    exact hx

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro n
  2. 0002intro hzero
  3. 0003intro hone
  4. 0004have hs : exists k. n=S k
  5. 0005specialize nonzero_is_succ (n)
  6. 0006apply nonzero_is_succ
  7. 0007exact hzero
  8. 0008cases hs
  9. 0009have hx : ~(x=0)
  10. 0010intro hxzero
  11. 0011apply hone
  12. 0012trans S x
  13. 0013exact hs_witness
  14. 0014rewrite hxzero
  15. 0015refl
  16. 0016rewrite hs_witness
  17. 0017specialize succ_le_succ (1)
  18. 0018specialize succ_le_succ (x)
  19. 0019apply succ_le_succ
  20. 0020specialize one_le_of_ne_zero (x)
  21. 0021apply one_le_of_ne_zero
  22. 0022exact hx