BA001D

cf_approximation_positive_coefficient_bound

A positive coefficient makes the nonnegative error contribution at least the current error, including error 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

∀ E. ∀ F. ∀ c. ∀ d. ¬c = 0 → Le(E,c · E + d · F)

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

Definition DAG

Le(a,b) · 1

Actual proof prerequisites

one_le_of_ne_zero · checked external prerequisitele_mul_of_one_le_left · checked external prerequisitele_add_right · checked external prerequisitele_trans · checked external prerequisite
Original expanded first-order statement 
forall E F c d. ~(c = 0) -> (exists cfba_bound_positive_coefficient. cfba_bound_positive_coefficient + (E) = (c * E + d * F))

Complete tactic proof in conservative notation

All 18 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

18 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 assumptionsL1–5

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

  1. L1
    intro E
  2. L2
    intro F
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro hc
02Use earlier factsL6–15

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

  1. L6
    specialize le_trans (E)
  2. L7
    specialize le_trans (c * E)
  3. L8
    specialize le_trans (c * E + d * F)
  4. L9
    apply le_trans
  5. L10
    specialize le_mul_of_one_le_left (c)
  6. L11
    specialize le_mul_of_one_le_left (E)
  7. L12
    apply le_mul_of_one_le_left
  8. L13
    specialize one_le_of_ne_zero (c)
  9. L14
    apply one_le_of_ne_zero
  10. L15
    exact hc
03Use earlier factsL16–18

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

  1. L16
    specialize le_add_right (c * E)
  2. L17
    specialize le_add_right (d * F)
  3. L18
    apply le_add_right

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro E
  2. 0002intro F
  3. 0003intro c
  4. 0004intro d
  5. 0005intro hc
  6. 0006specialize le_trans (E)
  7. 0007specialize le_trans (c * E)
  8. 0008specialize le_trans (c * E + d * F)
  9. 0009apply le_trans
  10. 0010specialize le_mul_of_one_le_left (c)
  11. 0011specialize le_mul_of_one_le_left (E)
  12. 0012apply le_mul_of_one_le_left
  13. 0013specialize one_le_of_ne_zero (c)
  14. 0014apply one_le_of_ne_zero
  15. 0015exact hc
  16. 0016specialize le_add_right (c * E)
  17. 0017specialize le_add_right (d * F)
  18. 0018apply le_add_right