BA000F

cf_approximation_opposite_errors_linear_absolute

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

Opposite signed errors combine without cancellation after subtracting the previous coefficient vector.

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 E F c d. (((a * v = b * u + E) /\ (b * U = a * V + F)) \/ ((b * u = a * v + E) /\ (a * V = b * U + F))) -> (((c * (a * v) + d * (b * U)) = (c * (b * u) + d * (a * V)) + (c * E + d * F)) \/ ((c * (b * u) + d * (a * V)) = (c * (a * v) + d * (b * U)) + (c * E + d * F)))

Constructive proof overview

Generated structural guide

Opposite signed errors combine without cancellation after subtracting the previous coefficient vector.

The unchanged tactic script uses 5 declared prerequisites and contains 22 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_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

22 script commands · 6 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 E
  8. L8
    intro F
  9. L9
    intro c
  10. L10
    intro d
02Fix variables and assumptionsL11–11

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

  1. L11
    intro h
03Separate the logical casesL12–14

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

  1. L12
    cases h
  2. L13
    cases h_left
  3. L14
    left
04Calculate and transport equalitiesL15–17

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

  1. L15
    rewrite h_left_left
  2. L16
    rewrite h_left_right
  3. L17
    simp [mul_add, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]
05Separate the logical casesL18–19

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

  1. L18
    cases h_right
  2. L19
    right
06Calculate and transport equalitiesL20–22

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

  1. L20
    rewrite h_right_left
  2. L21
    rewrite h_right_right
  3. L22
    simp [mul_add, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro u
  4. 0004intro U
  5. 0005intro v
  6. 0006intro V
  7. 0007intro E
  8. 0008intro F
  9. 0009intro c
  10. 0010intro d
  11. 0011intro h
  12. 0012cases h
  13. 0013cases h_left
  14. 0014left
  15. 0015rewrite h_left_left
  16. 0016rewrite h_left_right
  17. 0017simp [mul_add, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]
  18. 0018cases h_right
  19. 0019right
  20. 0020rewrite h_right_left
  21. 0021rewrite h_right_right
  22. 0022simp [mul_add, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail]