BA001A

cf_approximation_balance_exchange_columns

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

Exchanging the two genuine basis columns exchanges their coefficients and preserves the represented integer.

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 x y u U p n q m. ((x) + ((n) * (u) + (m) * (U)) = (y) + ((p) * (u) + (q) * (U))) -> ((x) + ((m) * (U) + (n) * (u)) = (y) + ((q) * (U) + (p) * (u)))

Constructive proof overview

Generated structural guide

Exchanging the two genuine basis columns exchanges their coefficients and preserves the represented integer.

The unchanged tactic script uses 1 declared prerequisite and contains 14 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_comm Stable theorem; checked-use authorized

Direct dependents

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

14 script commands · 4 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 assumptionsL1–9

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro u
  4. L4
    intro U
  5. L5
    intro p
  6. L6
    intro n
  7. L7
    intro q
  8. L8
    intro m
  9. L9
    intro h
02Calculate and transport equalitiesL10–12

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

  1. L10
    trans x + (n * u + m * U)
  2. L11
    simp [add_comm]
  3. L12
    trans y + (p * u + q * U)
03Use earlier factsL13–13

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

  1. L13
    exact h
04Calculate and transport equalitiesL14–14

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

  1. L14
    simp [add_comm]

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro u
  4. 0004intro U
  5. 0005intro p
  6. 0006intro n
  7. 0007intro q
  8. 0008intro m
  9. 0009intro h
  10. 0010trans x + (n * u + m * U)
  11. 0011simp [add_comm]
  12. 0012trans y + (p * u + q * U)
  13. 0013exact h
  14. 0014simp [add_comm]