BA001D

cf_approximation_positive_coefficient_bound

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

A positive coefficient makes the nonnegative error contribution at least the current error, including error 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 E F c d. ~(c = 0) -> (exists cfba_bound_positive_coefficient. cfba_bound_positive_coefficient + (E) = (c * E + d * F))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 4 declared prerequisites and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

one_le_of_ne_zero Stable theorem; checked-use authorized le_mul_of_one_le_left Alpha theorem; checked-use authorized le_add_right Stable theorem; checked-use authorized le_trans Stable theorem; checked-use authorized

Direct dependents

BA001E cf_approximation_small_positive_sum_current_zero BA0023 cf_approximation_subtractive_error_lower_bound

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

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.

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 exact 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