BA001C

cf_approximation_unimodular_signed_basis

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

Both determinant orientations represent every signed numerator and every natural denominator with actual integer coefficients.

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. (u * V + 1 = U * v \/ U * v + 1 = u * V) -> exists c d. (((((rp) = (rn) + ((c) * (u) + (d) * (U))) /\ ((t) = (0) + ((c) * (v) + (d) * (V))))) \/ (((((rp) + (d) * (U) = (rn) + (c) * (u)) /\ ((t) + (d) * (V) = (0) + (c) * (v)))) \/ (((((rp) + (c) * (u) = (rn) + (d) * (U)) /\ ((t) + (c) * (v) = (0) + (d) * (V)))) \/ ((((rp) + ((c) * (u) + (d) * (U)) = (rn)) /\ ((t) + ((c) * (v) + (d) * (V)) = (0)))))))

Constructive proof overview

Generated structural guide

Both determinant orientations represent every signed numerator and every natural denominator with actual integer coefficients.

The unchanged tactic script uses 2 declared prerequisites and contains 27 exact native proof lines.

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

Proof neighborhood

Direct dependencies

BA0019 cf_approximation_determinant_one_signed_basis BA001B cf_approximation_determinant_minus_one_signed_basis

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

27 script commands · 4 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.

Named ingredients (2)
cf approximation determinant one signed basiscf approximation determinant minus one signed basis
01Fix variables and assumptionsL1–8

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 hd
02Separate the logical casesL9–9

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

  1. L9
    cases hd
03Use earlier factsL10–19

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

  1. L10
    specialize cf_approximation_determinant_minus_one_signed_basis (u)
  2. L11
    specialize cf_approximation_determinant_minus_one_signed_basis (U)
  3. L12
    specialize cf_approximation_determinant_minus_one_signed_basis (v)
  4. L13
    specialize cf_approximation_determinant_minus_one_signed_basis (V)
  5. L14
    specialize cf_approximation_determinant_minus_one_signed_basis (rp)
  6. L15
    specialize cf_approximation_determinant_minus_one_signed_basis (rn)
  7. L16
    specialize cf_approximation_determinant_minus_one_signed_basis (t)
  8. L17
    apply cf_approximation_determinant_minus_one_signed_basis
  9. L18
    exact hd_left
  10. L19
    specialize cf_approximation_determinant_one_signed_basis (u)
04Use earlier factsL20–27

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

  1. L20
    specialize cf_approximation_determinant_one_signed_basis (U)
  2. L21
    specialize cf_approximation_determinant_one_signed_basis (v)
  3. L22
    specialize cf_approximation_determinant_one_signed_basis (V)
  4. L23
    specialize cf_approximation_determinant_one_signed_basis (rp)
  5. L24
    specialize cf_approximation_determinant_one_signed_basis (rn)
  6. L25
    specialize cf_approximation_determinant_one_signed_basis (t)
  7. L26
    apply cf_approximation_determinant_one_signed_basis
  8. L27
    exact hd_right

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro u
  2. 0002intro U
  3. 0003intro v
  4. 0004intro V
  5. 0005intro rp
  6. 0006intro rn
  7. 0007intro t
  8. 0008intro hd
  9. 0009cases hd
  10. 0010specialize cf_approximation_determinant_minus_one_signed_basis (u)
  11. 0011specialize cf_approximation_determinant_minus_one_signed_basis (U)
  12. 0012specialize cf_approximation_determinant_minus_one_signed_basis (v)
  13. 0013specialize cf_approximation_determinant_minus_one_signed_basis (V)
  14. 0014specialize cf_approximation_determinant_minus_one_signed_basis (rp)
  15. 0015specialize cf_approximation_determinant_minus_one_signed_basis (rn)
  16. 0016specialize cf_approximation_determinant_minus_one_signed_basis (t)
  17. 0017apply cf_approximation_determinant_minus_one_signed_basis
  18. 0018exact hd_left
  19. 0019specialize cf_approximation_determinant_one_signed_basis (u)
  20. 0020specialize cf_approximation_determinant_one_signed_basis (U)
  21. 0021specialize cf_approximation_determinant_one_signed_basis (v)
  22. 0022specialize cf_approximation_determinant_one_signed_basis (V)
  23. 0023specialize cf_approximation_determinant_one_signed_basis (rp)
  24. 0024specialize cf_approximation_determinant_one_signed_basis (rn)
  25. 0025specialize cf_approximation_determinant_one_signed_basis (t)
  26. 0026apply cf_approximation_determinant_one_signed_basis
  27. 0027exact hd_right