BA002B

cf_convergent_no_index_below_zero

No natural index or remainder is strictly below zero.

Alpha v34 checked-use · first admitted v29 · 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.

The initial 0/1 convergent is included: u is natural, not necessarily positive. Comparison denominators are strictly smaller and positive. Signed competitors are represented by an arbitrary difference rp−rn. Approximation inequalities are proved from the trace, never stored as assumptions in Convergent.

Exact theorem in conservative defined notation

∀ j. ¬Lt(j,0)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

add_eq_zero_right · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall j. (exists cfba_gap_no_index_zero. cfba_gap_no_index_zero + S (j) = (0)) -> false

Complete tactic proof in conservative notation

All 9 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

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.

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 assumptionsL1–2

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 defined 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