BA001F

cf_approximation_positive_difference_coefficient_nonzero

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

A strictly positive denominator in a difference of two nonnegative columns has a nonzero positive coefficient.

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 t v V c d. ~(t = 0) -> t + d * V = 0 + c * v -> ~(c = 0)

Constructive proof overview

Generated structural guide

A strictly positive denominator in a difference of two nonnegative columns has a nonzero positive coefficient.

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_eq_zero_left Stable theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized

Direct dependents

BA0021 cf_approximation_small_denominator_signed_coordinates

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

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.

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 exact 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]