BA0002

cf_approximation_prepend_determinant_backward

The reverse determinant orientation has the same actual quotient-matrix transition.

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. ∀ q. U · v + 1 = u · V → (q · u + v) · U + 1 = (q · U + V) · u

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

Definition DAG

none

Actual proof prerequisites

cf_approximation_prepend_determinant_forward
Original expanded first-order statement 
forall u U v V q. U * v + 1 = u * V -> (q * u + v) * U + 1 = (q * U + V) * u

Complete tactic proof in conservative notation

All 13 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

13 script commands · 2 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 (1)
cf approximation prepend determinant forward
01Fix variables and assumptionsL1–6

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 q
  6. L6
    intro hd
02Use earlier factsL7–13

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

  1. L7
    specialize cf_approximation_prepend_determinant_forward (U)
  2. L8
    specialize cf_approximation_prepend_determinant_forward (u)
  3. L9
    specialize cf_approximation_prepend_determinant_forward (V)
  4. L10
    specialize cf_approximation_prepend_determinant_forward (v)
  5. L11
    specialize cf_approximation_prepend_determinant_forward (q)
  6. L12
    apply cf_approximation_prepend_determinant_forward
  7. L13
    exact hd

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro u
  2. 0002intro U
  3. 0003intro v
  4. 0004intro V
  5. 0005intro q
  6. 0006intro hd
  7. 0007specialize cf_approximation_prepend_determinant_forward (U)
  8. 0008specialize cf_approximation_prepend_determinant_forward (u)
  9. 0009specialize cf_approximation_prepend_determinant_forward (V)
  10. 0010specialize cf_approximation_prepend_determinant_forward (v)
  11. 0011specialize cf_approximation_prepend_determinant_forward (q)
  12. 0012apply cf_approximation_prepend_determinant_forward
  13. 0013exact hd