BA002B

cf_convergent_no_index_below_zero

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

No natural index or remainder is strictly below zero.

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 j. (exists cfba_gap_no_index_zero. cfba_gap_no_index_zero + S (j) = (0)) -> false

Constructive proof overview

Generated structural guide

No natural index or remainder is strictly below zero.

The unchanged tactic script uses 2 declared prerequisites and contains 9 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_eq_zero_right Stable theorem; checked-use authorized succ_ne_zero Stable theorem; checked-use authorized

Direct dependents

BA0039 cf_convergent_matrix_empty_constructor

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 · 3 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 j
  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 factsL4–9

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

  1. L4
    specialize succ_ne_zero (j)
  2. L5
    apply succ_ne_zero
  3. L6
    specialize add_eq_zero_right (x)
  4. L7
    specialize add_eq_zero_right (S j)
  5. L8
    apply add_eq_zero_right
  6. L9
    exact h_witness

Library-wide reading audit

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