BA0001

cf_approximation_prepend_determinant_forward

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

Prepending a genuine quotient matrix reverses the determinant-one orientation.

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 q. u * V + 1 = U * v -> (q * U + V) * u + 1 = (q * u + v) * U

Constructive proof overview

Generated structural guide

Prepending a genuine quotient matrix reverses the determinant-one orientation.

The unchanged tactic script uses 4 declared prerequisites and contains 10 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_comm Stable theorem; checked-use authorized

Direct dependents

BA0002 cf_approximation_prepend_determinant_backward BA0006 cf_approximation_prepend_identity

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

10 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 assumptionsL1–6

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
02Calculate and transport equalitiesL7–10

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

  1. L7
    trans (q * U) * u + (u * V + 1)
  2. L8
    simp [add_mul, mul_assoc, mul_comm, add_comm]
  3. L9
    rewrite hd
  4. L10
    simp [add_mul, mul_assoc, mul_comm, add_comm]

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro u
  2. 0002intro U
  3. 0003intro v
  4. 0004intro V
  5. 0005intro q
  6. 0006intro hd
  7. 0007trans (q * U) * u + (u * V + 1)
  8. 0008simp [add_mul, mul_assoc, mul_comm, add_comm]
  9. 0009rewrite hd
  10. 0010simp [add_mul, mul_assoc, mul_comm, add_comm]