BA0016

cf_approximation_cofactor_numerator_balance

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

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

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 -> (rp) + ((V * rn + U * t) * (u) + (v * rp) * (U)) = (rn) + ((V * rp) * (u) + (u * t + v * rn) * (U))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 7 declared prerequisites and contains 15 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_mul Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized four_square_add_swap_right_tail Alpha theorem; checked-use authorized natural_mul_swap_right_tail Alpha theorem; checked-use authorized

Direct dependents

BA0019 cf_approximation_determinant_one_signed_basis BA001B cf_approximation_determinant_minus_one_signed_basis

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

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.

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 exact 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]