GF0071

gaussian_search_no_index_below_zero

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

The empty search interval has no index, proved in the original natural arithmetic.

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 i. ~(exists ge_gap_empty_search. ge_gap_empty_search + S (i) = (0))

Constructive proof overview

Generated structural guide

The empty search interval has no index, proved in the original natural arithmetic.

The unchanged tactic script uses 1 declared prerequisite and contains 9 exact native proof lines.

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

Proof neighborhood

Direct dependencies

succ_ne_zero Stable theorem; checked-use authorized

Direct dependents

GF0075 gaussian_factor_search_coordinate_row GF0076 gaussian_factor_search_coordinate_rectangle GF0086 gaussian_product_empty_exists GF0092 gaussian_unit_empty_factorization GF00A0 gaussian_product_replace_balance GF00A7 gaussian_factor_empty_matching GF00AF gaussian_irreducible_products_associate_unique

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

9 script commands · 5 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro i
  2. L2
    intro h
02Separate the logical casesL3–3

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

  1. L3
    cases h
03Use earlier factsL4–5

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

  1. L4
    specialize succ_ne_zero (x+i)
  2. L5
    apply succ_ne_zero
04Calculate and transport equalitiesL6–7

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

  1. L6
    trans x+S i
  2. L7
    symm
05Use earlier factsL8–9

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

  1. L8
    apply PA4
  2. L9
    exact h_witness

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro i
  2. 0002intro h
  3. 0003cases h
  4. 0004specialize succ_ne_zero (x+i)
  5. 0005apply succ_ne_zero
  6. 0006trans x+S i
  7. 0007symm
  8. 0008apply PA4
  9. 0009exact h_witness