BA000E

cf_approximation_derived_invariant_determinant

The determinant-one equation is a proved consequence of the actual-prefix invariant, not a field assumed by the computation relation.

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

∀ a. ∀ b. ∀ u. ∀ U. ∀ v. ∀ V. ConvergentErrorInvariant(a,b,u,U,v,V) → u · V + 1 = U · v ∨ U · v + 1 = u · V

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

Definition DAG

ConvergentErrorInvariant(a,b,u,U,v,V) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall a b u U v V. (exists cfba_error_derived_determinant cfba_previous_error_derived_determinant. (((((u * V + 1 = U * v) /\ ((a * v = b * u + cfba_error_derived_determinant) /\ (b * U = a * V + cfba_previous_error_derived_determinant)))) \/ (((U * v + 1 = u * V) /\ ((b * u = a * v + cfba_error_derived_determinant) /\ (a * V = b * U + cfba_previous_error_derived_determinant))))) /\ ((exists cfba_gap_derived_determinantdecrease. cfba_gap_derived_determinantdecrease + S (cfba_error_derived_determinant) = (cfba_previous_error_derived_determinant)) /\ (exists cfba_bound_derived_determinantprevious_bound. cfba_bound_derived_determinantprevious_bound + (cfba_previous_error_derived_determinant) = (b))))) -> (u * V + 1 = U * v \/ U * v + 1 = u * V)

Complete tactic proof in conservative notation

All 17 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

17 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 assumptionsL1–7

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro u
  4. L4
    intro U
  5. L5
    intro v
  6. L6
    intro V
  7. L7
    intro hi
02Separate the logical casesL8–13

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

  1. L8
    cases hi
  2. L9
    cases hi_witness
  3. L10
    cases hi_witness_witness
  4. L11
    cases hi_witness_witness_left
  5. L12
    cases hi_witness_witness_left_left
  6. L13
    left
03Use earlier factsL14–14

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

  1. L14
    exact hi_witness_witness_left_left_left
04Separate the logical casesL15–16

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

  1. L15
    cases hi_witness_witness_left_right
  2. L16
    right
05Use earlier factsL17–17

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

  1. L17
    exact hi_witness_witness_left_right_left

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro u
  4. 0004intro U
  5. 0005intro v
  6. 0006intro V
  7. 0007intro hi
  8. 0008cases hi
  9. 0009cases hi_witness
  10. 0010cases hi_witness_witness
  11. 0011cases hi_witness_witness_left
  12. 0012cases hi_witness_witness_left_left
  13. 0013left
  14. 0014exact hi_witness_witness_left_left_left
  15. 0015cases hi_witness_witness_left_right
  16. 0016right
  17. 0017exact hi_witness_witness_left_right_left