BA0010

cf_approximation_subtract_previous_error_balance

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

Actual numerator and denominator coordinate equations transport the represented rational error exactly.

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 a b u U v V rp rn t c d. (((rp + d * U = rn + c * u) /\ (t + d * V = c * v))) -> (a * t + b * rn) + (c * (b * u) + d * (a * V)) = (c * (a * v) + d * (b * U)) + (b * rp)

Constructive proof overview

Generated structural guide

Actual numerator and denominator coordinate equations transport the represented rational error exactly.

The unchanged tactic script uses 6 declared prerequisites and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mul_add 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

Direct dependents

BA0011 cf_approximation_subtract_previous_absolute_error

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

18 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 assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro u
  4. L4
    intro U
  5. L5
    intro v
  6. L6
    intro V
  7. L7
    intro rp
  8. L8
    intro rn
  9. L9
    intro t
  10. L10
    intro c
02Fix variables and assumptionsL11–12

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

  1. L11
    intro d
  2. L12
    intro h
03Separate the logical casesL13–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L13
    cases h
04Calculate and transport equalitiesL14–18

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

  1. L14
    trans a * (t + d * V) + b * (rn + c * u)
  2. L15
    simp [mul_add, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]
  3. L16
    rewrite h_right
  4. L17
    rewrite <- h_left
  5. L18
    simp [mul_add, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro u
  4. 0004intro U
  5. 0005intro v
  6. 0006intro V
  7. 0007intro rp
  8. 0008intro rn
  9. 0009intro t
  10. 0010intro c
  11. 0011intro d
  12. 0012intro h
  13. 0013cases h
  14. 0014trans a * (t + d * V) + b * (rn + c * u)
  15. 0015simp [mul_add, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]
  16. 0016rewrite h_right
  17. 0017rewrite <- h_left
  18. 0018simp [mul_add, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]