BA001C

cf_approximation_unimodular_signed_basis

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

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

∀ u. ∀ U. ∀ v. ∀ V. ∀ rp. ∀ rn. ∀ t. u · V + 1 = U · v ∨ U · v + 1 = u · V → ∃ x. ∃ y. rp = rn + (x · u + y · U) ∧ t = 0 + (x · v + y · V) ∨ (rp + y · U = rn + x · u ∧ t + y · V = 0 + x · v ∨ (rp + x · u = rn + y · U ∧ t + x · v = 0 + y · V ∨ rp + (x · u + y · U) = rn ∧ t + (x · v + y · V) = 0))

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

Definition DAG

none

Actual proof prerequisites

cf_approximation_determinant_one_signed_basiscf_approximation_determinant_minus_one_signed_basis
Original expanded first-order 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)))))))

Complete tactic proof in conservative notation

All 27 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

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.

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 (2)
cf approximation determinant minus one signed basiscf approximation determinant 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 defined 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