GF0072

gaussian_search_two_le_nonzero_not_one

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

Alpha v34 checked-use · first admitted v30 · 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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

∀ n. ¬n = 0 → ¬n = 1 → Lt(1,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 prerequisiteone_le_of_ne_zero · checked external prerequisitesucc_le_succ · checked external prerequisite
Original expanded first-order statement 
forall n. ~(n=0) -> ~(n=1) -> (exists ge_gap_nonunit_norm_two. ge_gap_nonunit_norm_two + (2) = (n))

Complete tactic proof in conservative notation

All 22 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

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.

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 assumptionsL1–3

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 defined 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