BA001F

cf_approximation_positive_difference_coefficient_nonzero

A strictly positive denominator in a difference of two nonnegative columns has a nonzero positive 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 → t + d · V = 0 + c · v → ¬c = 0

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

Definition DAG

none

Actual proof prerequisites

add_eq_zero_left · checked external prerequisitemul_zero_left · checked external prerequisite
Original expanded first-order statement 
forall t v V c d. ~(t = 0) -> t + d * V = 0 + c * v -> ~(c = 0)

Complete tactic proof in conservative notation

All 16 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

16 script commands · 5 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 assumptionsL1–8

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 h
  8. L8
    intro hc
02Use earlier factsL9–12

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

  1. L9
    apply ht
  2. L10
    specialize add_eq_zero_left (t)
  3. L11
    specialize add_eq_zero_left (d * V)
  4. L12
    apply add_eq_zero_left
03Calculate and transport equalitiesL13–13

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

  1. L13
    trans 0 + c * v
04Use earlier factsL14–14

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

  1. L14
    exact h
05Calculate and transport equalitiesL15–16

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

  1. L15
    rewrite hc
  2. L16
    simp [mul_zero_left]

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro t
  2. 0002intro v
  3. 0003intro V
  4. 0004intro c
  5. 0005intro d
  6. 0006intro ht
  7. 0007intro h
  8. 0008intro hc
  9. 0009apply ht
  10. 0010specialize add_eq_zero_left (t)
  11. 0011specialize add_eq_zero_left (d * V)
  12. 0012apply add_eq_zero_left
  13. 0013trans 0 + c * v
  14. 0014exact h
  15. 0015rewrite hc
  16. 0016simp [mul_zero_left]