BA0020

cf_approximation_zero_current_sum_as_difference

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

The only small-denominator nonnegative combination is an actual previous-column difference with zero current 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 u U v V rp rn t c d. ((((rp) = (rn) + ((c) * (u) + (d) * (U))) /\ ((t) = (0) + ((c) * (v) + (d) * (V))))) -> c = 0 -> ((((rp) + (0) * (u) = (rn) + (d) * (U)) /\ ((t) + (0) * (v) = (0) + (d) * (V))))

Constructive proof overview

Generated structural guide

The only small-denominator nonnegative combination is an actual previous-column difference with zero current coefficient.

The unchanged tactic script uses 2 declared prerequisites and contains 25 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mul_zero_left Stable theorem; checked-use authorized zero_add 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

25 script commands · 8 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 assumptionsL1–10

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

  1. L1
    intro u
  2. L2
    intro U
  3. L3
    intro v
  4. L4
    intro V
  5. L5
    intro rp
  6. L6
    intro rn
  7. L7
    intro t
  8. L8
    intro c
  9. L9
    intro d
  10. L10
    intro h
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hc
03Separate the logical casesL12–13

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

  1. L12
    cases h
  2. L13
    split
04Calculate and transport equalitiesL14–16

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

  1. L14
    trans rp
  2. L15
    simp [mul_zero_left]
  3. L16
    trans rn + (c * u + d * U)
05Use earlier factsL17–17

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

  1. L17
    exact h_left
06Calculate and transport equalitiesL18–22

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

  1. L18
    rewrite hc
  2. L19
    simp [mul_zero_left, zero_add]
  3. L20
    trans t
  4. L21
    simp [mul_zero_left]
  5. L22
    trans 0 + (c * v + d * V)
07Use earlier factsL23–23

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

  1. L23
    exact h_right
08Calculate and transport equalitiesL24–25

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

  1. L24
    rewrite hc
  2. L25
    simp [mul_zero_left, zero_add]

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro u
  2. 0002intro U
  3. 0003intro v
  4. 0004intro V
  5. 0005intro rp
  6. 0006intro rn
  7. 0007intro t
  8. 0008intro c
  9. 0009intro d
  10. 0010intro h
  11. 0011intro hc
  12. 0012cases h
  13. 0013split
  14. 0014trans rp
  15. 0015simp [mul_zero_left]
  16. 0016trans rn + (c * u + d * U)
  17. 0017exact h_left
  18. 0018rewrite hc
  19. 0019simp [mul_zero_left, zero_add]
  20. 0020trans t
  21. 0021simp [mul_zero_left]
  22. 0022trans 0 + (c * v + d * V)
  23. 0023exact h_right
  24. 0024rewrite hc
  25. 0025simp [mul_zero_left, zero_add]