BA001E

cf_approximation_small_positive_sum_current_zero

A nonnegative combination with denominator below the current denominator has zero current coefficient.

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

∀ t. ∀ v. ∀ V. ∀ c. ∀ d. t = 0 + (c · v + d · V) → Lt(t,v) → c = 0

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

Definition DAG

Lt(a,b) · 1

Actual proof prerequisites

eq_decidable · checked external prerequisitecf_approximation_positive_coefficient_boundle_trans · checked external prerequisitelt_not_le · checked external prerequisitezero_add · checked external prerequisite
Original expanded first-order statement 
forall t v V c d. t = 0 + (c * v + d * V) -> (exists cfba_gap_small_positive_sum. cfba_gap_small_positive_sum + S (t) = (v)) -> c = 0

Complete tactic proof in conservative notation

All 35 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

35 script commands · 12 reading checkpoints · 1 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.

Named ingredients (1)
cf approximation positive coefficient bound
01Fix variables and assumptionsL1–7

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

  1. L1
    intro t
  2. L2
    intro v
  3. L3
    intro V
  4. L4
    intro c
  5. L5
    intro d
  6. L6
    intro ht
  7. L7
    intro hlt
02Establish hcL8–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eq decidable.

  1. L8
    have hc : c = 0 \/ ~(c = 0)
  2. L9
    specialize eq_decidable (c)
  3. L10
    specialize eq_decidable (0)
  4. L11
    apply eq_decidable
03Separate the logical casesL12–12

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

  1. L12
    cases hc
04Use earlier factsL13–13

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

  1. L13
    exact hc_left
05Separate the logical casesL14–14

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

  1. L14
    exfalso
06Use earlier factsL15–24

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

  1. L15
    specialize lt_not_le (t)
  2. L16
    specialize lt_not_le (v)
  3. L17
    apply lt_not_le
  4. L18
    exact hlt
  5. L19
    specialize le_trans (v)
  6. L20
    specialize le_trans (c * v + d * V)
  7. L21
    specialize le_trans (t)
  8. L22
    apply le_trans
  9. L23
    specialize cf_approximation_positive_coefficient_bound (v)
  10. L24
    specialize cf_approximation_positive_coefficient_bound (V)
07Use earlier factsL25–28

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

  1. L25
    specialize cf_approximation_positive_coefficient_bound (c)
  2. L26
    specialize cf_approximation_positive_coefficient_bound (d)
  3. L27
    apply cf_approximation_positive_coefficient_bound
  4. L28
    exact hc_right
08Construct an explicit witnessL29–29

Supply the displayed value, then prove that it has the required property.

  1. L29
    exists 0
09Calculate and transport equalitiesL30–30

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

  1. L30
    trans c * v + d * V
10Use earlier factsL31–31

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

  1. L31
    apply zero_add
11Calculate and transport equalitiesL32–33

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

  1. L32
    symm
  2. L33
    trans 0 + (c * v + d * V)
12Use earlier factsL34–35

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

  1. L34
    exact ht
  2. L35
    apply zero_add

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro t
  2. 0002intro v
  3. 0003intro V
  4. 0004intro c
  5. 0005intro d
  6. 0006intro ht
  7. 0007intro hlt
  8. 0008have hc : c = 0 \/ ~(c = 0)
  9. 0009specialize eq_decidable (c)
  10. 0010specialize eq_decidable (0)
  11. 0011apply eq_decidable
  12. 0012cases hc
  13. 0013exact hc_left
  14. 0014exfalso
  15. 0015specialize lt_not_le (t)
  16. 0016specialize lt_not_le (v)
  17. 0017apply lt_not_le
  18. 0018exact hlt
  19. 0019specialize le_trans (v)
  20. 0020specialize le_trans (c * v + d * V)
  21. 0021specialize le_trans (t)
  22. 0022apply le_trans
  23. 0023specialize cf_approximation_positive_coefficient_bound (v)
  24. 0024specialize cf_approximation_positive_coefficient_bound (V)
  25. 0025specialize cf_approximation_positive_coefficient_bound (c)
  26. 0026specialize cf_approximation_positive_coefficient_bound (d)
  27. 0027apply cf_approximation_positive_coefficient_bound
  28. 0028exact hc_right
  29. 0029exists 0
  30. 0030trans c * v + d * V
  31. 0031apply zero_add
  32. 0032symm
  33. 0033trans 0 + (c * v + d * V)
  34. 0034exact ht
  35. 0035apply zero_add