BA0016

cf_approximation_cofactor_numerator_balance

The determinant-one cofactor formula reconstructs every actual signed numerator, not merely positive vectors.

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 → rp + ((V · rn + U · t) · u + v · rp · U) = rn + (V · rp · u + (u · t + v · rn) · U)

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

Definition DAG

none

Actual proof prerequisites

add_mul · checked external prerequisitemul_assoc · checked external prerequisitemul_comm · checked external prerequisiteadd_assoc · checked external prerequisiteadd_comm · checked external prerequisitefour_square_add_swap_right_tail · checked external prerequisitenatural_mul_swap_right_tail · checked external prerequisite
Original expanded first-order statement 
forall u U v V rp rn t. U * v + 1 = u * V -> (rp) + ((V * rn + U * t) * (u) + (v * rp) * (U)) = (rn) + ((V * rp) * (u) + (u * t + v * rn) * (U))

Complete tactic proof in conservative notation

All 15 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

15 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.

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
02Calculate and transport equalitiesL9–15

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

  1. L9
    trans (u * V) * rn + ((U * v + 1) * rp + (u * U) * t)
  2. L10
    simp [add_mul, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]
  3. L11
    trans (U * v + 1) * rn + ((u * V) * rp + (u * U) * t)
  4. L12
    rewrite hd
  5. L13
    rewrite hd
  6. L14
    refl
  7. L15
    simp [add_mul, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]

Library-wide reading audit

Original defined command ledger · 15 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. 0009trans (u * V) * rn + ((U * v + 1) * rp + (u * U) * t)
  10. 0010simp [add_mul, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]
  11. 0011trans (U * v + 1) * rn + ((u * V) * rp + (u * U) * t)
  12. 0012rewrite hd
  13. 0013rewrite hd
  14. 0014refl
  15. 0015simp [add_mul, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]